Abstract
Two-dimensional direction-of-arrival (DOA) estimation in coprime planar array involves problems that the complexity of spectral peak search is huge and the noncircular feature of signals is not considered. Considering that unitary estimating signal parameters via rotational invariance techniques (Unitary-ESPRIT) is a low complexity subspace algorithm, an approach to estimate DOA fast for multiple signals is proposed in this paper. We first apply Unitary-ESPRIT to solve one possible value of each signal. Given the relationship between the ambiguous values and real values, we then have all possible values belonging to each subarray. Through finding the common values of two subarrays, we finally obtain the highly precise true DOAs. Moreover, when the signals are noncircular, we present an improved method using noncircular Unitary-ESPRIT, which is favorable in terms of accuracy and degree of freedom. Simulation results demonstrate the effectiveness of our proposed methods.
1. Introduction
Direction-of-arrival (DOA) estimation of multiple narrowband signals is a fundamental task in many applications such as radar [2], underwater acoustics [3, 4], indoor navigation [5, 6], and wireless communication [7]. The uniform rectangular arrays (URAs) [8] using subspace algorithms can obtain high-resolution two-dimensional (2D) DOA estimations. In recent years, the coprime array [9–12] has become a focus, which consists of two uniform sparse arrays. Compared with URAs, the coprime planar array has a higher array aperture with the same number of sensors. And a partial spectrum search (PSS) method in coprime planar array is presented in [13, 14], which resolves the DOAs via finding the common peaks of two subarrays, but the spectral peak search costs much complexity, especially with a small searching step to acquire high accuracy. We have proposed a method in [1], which uses the covariance matrix of coprime planar array to estimate a new covariance matrix with matrix completion theory. This processing has much improved degree of freedom (DOF) bigger than the number of sensors, but the matrix completion costs much complexity and introduces the additional errors. Thus, when we want to secure a highly accurate estimation value with small complexity, this method may be not efficient enough.
Furthermore, the existing algorithms have just taken the independent narrowband signals into consideration. But sometimes the features of impinging signals can influence the performance. Considering that the general subspace methods cannot be applied to coherent signals, we have proposed an algorithm in [15] to resolve DOAs of both uncorrelated and coherent signals in coprime planar array. Moreover, there are many communication signals with ASK, BPSK, and UQPSK modulation [16, 17], which have the noncircular features, and some with QPSK and QAM modulation [18], which have the circular features. Independent noncircular signals, whose ellipse covariance matrix is nonzero, can expand the array aperture and enhance the accuracy [19, 20], while the circular signals cannot. However, the extended array aperture causes the complexity increased. As a result, it is meaningful that we additionally propose a low complexity method for noncircular signals, which can utilize the noncircular feature to realize fast DOA estimation in coprime planar array. An algorithm is presented for noncircular signals [21], but it can only have favorable performance when array aperture is not big.
In order to avoid the high complexity cost by the spectral searching and utilize the signals features, in this paper, a fast DOA estimation method in coprime array for multiple signals is proposed. We first obtain the received signal data and calculate the covariance matrix. Then, when signals are circular, we can apply unitary estimating signal parameters via rotational invariance techniques (Unitary-ESPRIT) [22–25], a low complexity algorithm, to solve the closed-form solutions, which are possible estimated values. When the signals are noncircular, we make an improvement and present the noncircular Unitary-ESPRIT (NC-Unitary-ESPRIT) to solve the closed-form solutions. Through the relationship between the ambiguous values and real values, we can obtain all the possible values of each subarray. By finding the common values between two subarrays, the true estimated DOAs are obtained with high precision. Compared with PSS, both Unitary-ESPRIT and NC-Unitary-ESPRIT can reduce the complexity. Moreover, NC-Unitary-ESPRIT has much improved the accuracy and DOF with the noncircular feature.
The paper is organized as follows. We first present a brief signal model in Section 2. In Section 3, we explain our approach to estimate DOAs and show the steps of the proposed algorithm. The computational complexity and DOF analysis are presented in Section 4. In Section 5, we show the results of simulations. Finally, we summarize work in Section 6. Throughout this paper, represents the dimensional unit array; and denote transpose, and Hermitian transpose, respectively; represents a vector transforming to a diagonal matrix.
2. Signal Model
Considering the coprime planar array [1], the model is made up of two URAs shown in Figure 1. Subarray 1 has sensors and subarray 2 has sensors, where and are coprime integers (assuming ). The th () subarray sets the space distance between the two adjacent sensors as ( and ), where is the wavelength of impinging signals. The two subarrays coincide at the origin, so the total number of sensors is . Suppose that there are narrowband far-field independent signals impinging on the array with the power . The th signal comes from the elevation angle and azimuth angle . Hence, the received signals at the th subarray can be represented asThe array manifold matrix is denoted aswhere is a steering vector of size and the element in the ()th ( row is represented as . The signal data vector denotes the sampling time, where and is the number of snapshots. And the noise vectorwhere is usually Gaussian random variables with zero means and variance .
We then can calculate the covariance matrix of received signals given bywhere . Apply 2D multiple signal classification (2D-MUSIC) [8, 26] to this covariance matrix and obtain the spatial spectrum function expressed aswhere is the noise subspace of the th subarray. Because the distance of adjacent sensors in each subarray is over half wavelength of impinging signals, through finding the peaks of spectrum, we can obtain the both ambiguous angles and real angles. But the common values of the estimated values of two subarrays are the real values.
3. Fast DOA Estimation of Multiple Signals
Unitary-ESPRIT is an efficient method, which solves the closed-form solutions of estimated values and reduces much complexity compared with the spectral peak search method, such as 2D-MUSIC. Unfortunately, Unitary-ESPRIT is commonly used in nonsparse arrays and it cannot estimate all ambiguous and real values.
3.1. Relationship between the Ambiguous Values and Real Values
Assume that there is a signal from the direction and one estimated ambiguous value is . Hence, the relationship between the real angle and one ambiguous angle of the th subarray can be given bySince and , we have and . Thus, we haveand and are the integers in the ranges of . As a result, when we can obtain one estimation angle , no matter the real angle or one ambiguous angle, we can obtainwhere and are the sets with values containing the unique real value. Combine all the sets and defineThrough finding common values between and , and , we can finally select the corresponding true value as .
3.2. Unitary-ESPRIT for DOA Estimation
We can use Unitary-ESPRIT to obtain one possible estimated value of each signal. First, we define as a inverse matrix of size , whose counter-diagonal values are 1 while the others are 0. Thus, the unitary matrix satisfies . For example, the unitary matrix of odd order is expressed asMoreover, the matrix of even order is obtained from (11) by dropping its center row and center column. We then transform the complex-value covariance matrix to real-value covariance matrix asNext, we define the selection matrices and and construct the desired dimensional selection matrices as and , where represents the Kronecker product. Therefore, we definewhere and represent the real part and imaginary part of the matrix or vector, respectively. Take eigenvalue decomposition of and obtain the signal subspace . At last, we combine the last two steps and havewhere is denoted as pseudoinverse operator. Define , and calculate its eigenvalue vector . We can acquire the values and , which are denoted asHence, considering (9) and (10), we can remove all ambiguous values and obtain the corresponding true values . The final estimated DOAs are defined as
3.3. DOA Estimation for Noncircular Signals
The last subsection presents the algorithm for circular signals, which can also be used for general independent signals. When there are noncircular signals, the ellipse covariance matrix of them can be nonzero; thus, we need add some additional procession. We assume that the signals are strictly noncircular. At first, we can “double” the available sensors byLet us definewhere can be a real-value matrix. The covariance matrix can be given byNext, the selection matrices should be modified in the following fashion:We then haveTake eigenvalue decomposition of and obtain the signal subspace . We combine the last two steps and finally haveDefine , and calculate its eigenvalue vector . We can acquire the values and . Similarly, considering (9) and (10), we can obtain the corresponding true values . And the estimation can be calculated via (20).
3.4. Algorithm Steps Conclusion
A detailed flowchart of this algorithm is shown in Figure 2. The main steps of the proposed algorithm can be summarized as follows:
Step 1. Calculate the covariance matrix via (5).
Step 2. Apply Unitary-ESPRIT and obtain the estimation .
Step 3. If the signals are noncircular, we transform the received signal data as (22) and then apply NC-Unitary-ESPRIT to obtain the estimation .
Step 4. Calculate all real values and ambiguous values through (9) and (10) and find the corresponding real values as or .
Step 5. Resolve the DOA estimation via (20).
4. Analysis of Computational Complexity and DOF
4.1. Analysis of Computational Complexity
We first analyze the computational complexity of the proposed method using Unitary-ESPRIT algorithm and compare it with the spectral peak search method based on 2D-MUSIC algorithm and PSS. And then analyze the complexity of proposed method using NC-Unitary-ESPRIT when signals are noncircular.
The complexity of the proposed method using Unitary-ESPRIT mainly concludes three parts. The complexities of calculating the covariance matrix, eigenvalue decomposition, and Unitary-ESPRIT are , , and , respectively. Thus, the total complexity of proposed algorithm using Unitary-ESPRIT is . And 2D-MUSIC and PSS cost the complexities as and , respectively, where and denote the spectral points. Similarly, the complexity of the proposed method using NC-Unitary-ESPRIT also has three parts: calculating the covariance matrix as , eigenvalue decomposition as , and NC-Unitary-ESPRIT as . Hence, the total complexity of proposed algorithm using NC-Unitary-ESPRIT is . For the sake of clarity, the computational complexities of all these approaches are summarized in Table 1. Moreover, we compare the complexities of methods versus the number of sensors and the searching step (, where ) in Figures 3(a) and 3(b), respectively.
(a)
(b)
As shown in Figure 3, compared with the spectral peak search methods, the proposed algorithms using Unitary-ESPRIT and NC-Unitary-ESPRIT have reduced much complexity. The searching step has an impact on the spectral peak search methods but no influence to proposed algorithms. Hence, the proposed algorithms can obtain the high accuracy while the spectral peak search methods need a small searching step and cost more complexity to realize that. Moreover, because the method for noncircular signals expands the sensors, the complexity of NC-Unitary-ESPRIT is higher than Unitary-ESPRIT, but the gap is not big. We can obtain a higher precise estimation of noncircular signals with a little complexity increase.
4.2. Analysis of DOF
The DOF determines the maximum number of signals that we can estimate directly. 2D-MUSIC, PSS and the proposed algorithms are all common values finding method to resolve the DOAs, where the DOFs of those algorithms are limited by the number of subarray sensors. The DOFs of 2D-MUSIC, PSS, and proposed algorithm using Unitary-ESPRIT are equal as . When the signals are noncircular, because NC-Unitary-ESPRIT can “double” the sensors, the DOF of this method is . The detailed values are presented in Figure 4. The figure shows that NC-Unitary-ESPRIT does increase the DOF compared with the Unitary-ESPRIT. But DOFs of the proposed algorithms are both smaller than the number of sensors, which means that DOF may be further improved with one proper method.
5. Simulation Results
This section performs the results of simulation experiments comparing the two proposed methods with PSS. To measure the accuracy of the algorithms, define the root mean square error (RMSE) aswhere , , and are the number of Monte Carlo simulations, the th real values and the estimated values, respectively. We assume that the impinging noncircular signals are BPSK modulation. And if the impinging signals are circular signals, they are QPSK modulation.
(1) Feasibility Demonstration of Proposed Algorithms. We assume that there is one impinging signal from . We use Unitary-ESPRIT to resolve the DOA and the results of each step are listed in Table 2. Obviously, , and then . Hence, we prove that the proposed algorithm can estimate DOA without the spectral peak search.
(2) Performance of Estimating Multiple Signals. The two proposed methods can estimate multiple signals. To verify this ability, we set and () and consider the case of circular signals and noncircular signals with a signal-of-noise ratio (SNR) of dB and , respectively. The distribution of estimated values from 100 simulations using two proposed methods is presented in Figures 5(a) and 5(b), respectively. The figure demonstrates that the proposed algorithms can completely estimate DOAs of multiple signals. Moreover, the estimated values are all concentrated around the real values.
(a) = 7 circular signals
(b) = 16 noncircular signals
(3) RMSE Comparison under Different SNRs. In this simulation, we study the RMSE performance of proposed algorithms under different SNRs in the setting with , and SNRs from dB to dB at dB intervals, compared with PSS. We set that there are signals which are strictly noncircular, where both Unitary-ESPRIT and NC-Unitary-ESPRIT can be applied. The results are shown in Figure 6. We can see that the RMSE of Unitary-ESPRIT is smaller than that of PSS with the searching step , and the gap between them become bigger with the increase of SNR. However, when the searching step of PSS is , the RMSE of it is much smaller. But considering the complexity, that of PSS is increased from to while that of Unitary-ESPRIT is only , which means that PSS needs to cost much complexity to improve the accuracy. When the signals are noncircular, we apply the NC-Unitary-ESPRIT with the cost of complexity as and obtain the highest accuracy, which is computationally efficient and favorable.
(a)
(b)
In addition, another advantage of NC-Unitary-ESPRIT is that it can estimate more signals than PSS and Unitary-ESPRIT. Hence, we set four situations where the number of noncircular signals is , , , and , respectively. The RMSEs of those different situations are presented in Figure 7. When the number of signals is bigger than that of subarray sensors, it can still maintain the high accuracy.
(a)
(b)
(4) RMSE Comparison under Different Number of Snapshots. Consider situations used in simulation (3) again. The number of snapshots is varied in the range from , when , and the results are shown in Figure 8. With the increase of number of snapshots, the accuracy of the proposed algorithms can be higher. Moreover, the decline gradually reaches a plateau. The other conclusions are the same as those from simulation (3).
(a)
(b)
6. Conclusions
The paper has proposed a fast DOA estimation approach in a coprime planar array for multiple signals, where circular and noncircular signals are concerned. The paper has described the model and the associated algorithms and analyzed the computational complexity as well as DOF of the proposed methods in comparison with that of existing algorithms. Through the theoretical analysis and simulation experiments, we demonstrate that proposed methods can detect multiple signals and realize the estimation of DOAs. Unitary-ESPRIT can obtain a close accuracy as the spectral peak search methods. Although the spectral peak search methods can obtain a higher accuracy with a small searching step, it costs much larger complexity than Unitary-ESPRIT. Moreover, when we estimate the DOAs of strictly noncircular signals, NC-Unitary-ESPRIT is favorable in both accuracy and DOF compared with existing algorithms and Unitary-ESPRIT. In practice, there can be the coexistence of both circular and noncircular signals. Hence, how to separate the two kinds of signals and resolve their DOAs with the consideration of signals features deserves the further research.
Data Availability
The data, which are produced by simulations, used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work is supported by the National Natural Science Foundation of China (Grant no. 61401513).