A high-precision 2D DOA estimation algorithm by a large-space three-parallel array for reducing mutual coupling☆
Introduction
Direction of arrival (DOA) estimation of multiple spatial signals by array antenna has significant meaning in the fields such as mobile communication [1], electronic warfare [2], and so on. In order to get accurate spatial orientation information of signal sources, we should obtain the estimation of the elevation angles and azimuth angles of received signals simultaneously, which is the so-called two-dimensional (2D) DOA estimation. In recent years, many high-performance 2D DOA estimation algorithms based on different kinds of planar arrays such as L-shaped array [3], [4], rectangular array [5] and uniform circular array [6] have been proposed. Multiple-parallel linear array [7], [8], [9], [10], [11], [12], [13], [14] is another frequently-used planar array composed of multiple parallel linear arrays. Compared with rectangular array, multiple-parallel linear array has more generalized array architecture.
In [7], authors used the covariance of the received data from two parallel linear arrays to construct a DOA matrix, by eigenvalue decomposition (EVD) of which the self-matching elevation angles and azimuth angles can be estimated. An extended DOA matrix algorithm was proposed in [8], and it can improve the precision of traditional DOA matrix algorithm by making full use of the correlation information. In [9], authors proposed a polynomial root-finding-based algorithm for 2D DOA estimation by using a two-parallel linear array. In addition to EVD, the process of polynomial root-finding also involves high complexity. The rank-reduction algorithm [10] is based on a three-parallel array, and it can estimate the 2D angles by one-dimensional (1D) search. However, as [7], [8], [9], the EVD of the covariance matrix is still an essential process for this algorithm.
Propagator method (PM) [11], [12], [13], [14], [15]can obtain a propagator matrix without EVD of covariance matrix, and the propagator matrix can be used to obtain a signal subspace or noise subspace. In [11], authors have proposed a 2D PM algorithm to estimate the elevation and azimuth angles by using a two-parallel linear array. In [12], authors have improved the algorithm [11], and proposed a new 2D PM algorithm with higher estimation precision and lower complexity. Not only that, for the 2D PM algorithm [12], the elevation angles and azimuth angles can be matched automatically. For reducing the complexity of 2D DOA estimation, an array manifold matching (AMM) algorithm [13] was proposed, and it has been used with PM algorithm to estimate the 2D DOA. Although the complexity is further reduced, the estimation accuracy of this algorithm is lower than PM algorithms [11], [12]. Nevertheless, all the algorithms [7], [8], [9], [10], [11], [12], [13] are based on an assumption that mutual coupling has been ignored. In practical applications, the performance of array will be affected by mutual coupling, particularly for the small-spacing array. In [14], a 2D PM algorithm based on a large-space two-parallel array was proposed, which can reduce the effect of mutual coupling to a certain extent. But the effect of mutual coupling still can't be eliminated completely.
In fact, many sparse arrays [16], [17], [18], [19] have the effect of reducing mutual coupling. But in order to avoid angle ambiguity, the minimum internal spacing still can't exceed half-wavelength of the signal. Hence, mutual coupling is difficult to be eliminated completely. In [20] and [21], an unfolded parallel co-prime array and a generalized co-prime nested array were proposed, respectively. For the two kinds of array, minimum internal spacing can exceed half-wavelength of the signal, but they only work for some special algorithms. There are many self-correcting algorithms [22], [23], [24], [25], [26] to remove the effect of mutual coupling, while these methods may lead to loss of degrees of freedom and performance degradation.
In this paper, we first propose a new three-parallel linear array configuration. The element-spacing of the middle sub-array and the other two sub-arrays are h units and h+1 units, respectively. The spacings of adjacent sub-arrays are p units and p+1 units, respectively. Theoretically, both h and p can be any positive integer. Hence, the mutual coupling in the array can be neglected when h and p are large enough. Then, an extended PM algorithm is introduced according to the special array. The main contributions of the proposed scheme are as follows:
- 1.
the proposed array is robust to the mutual coupling effect, which has practical significance;
- 2.
the proposed algorithm can eliminate angle ambiguity;
- 3.
the estimation precision of the proposed algorithm is far higher than the existing 2D PM algorithms [11], [12], [13], [14].
Notation: In the paper, symbols , , and ⊗ stand for transpose, Hermitian transpose, Moore-Penrose pseudo-inverse, and Kronecker product respectively. The matrix composed of the i-th row to the j-th row of the matrix M is denoted by . The principal argument operation and expectation operation are denoted by and , respectively.
Section snippets
Array received model
Suppose that three parallel uniform linear arrays are located on plane as shown in Fig. 1. The construction of the three-parallel array is shown in Fig. 2. The number of sensors for the three-parallel array is , where . The middle array consisting of N sensors is located on z axis. The three-dimensional coordinates of the N sensors are where , and λ is the wavelength of signal. Two same M-element linear arrays are located on the two sides
Algorithm description
In this section, we introduce an extended PM algorithm according to the three-parallel array with mutual coupling being ignored. As PM algorithms [11], [12], [13], [14], we part matrix C as where and . It's easy to know that there must be a matrix satisfying
When noise is ignored, the covariance matrix can be expressed as where R can be estimated by .
Being similar to (7), we partition the
Cramer-Rao bound
As [12], [13], the Cramer-Rao Bound (CRB) of 2D DOA estimation with the proposed three-parallel linear array can be given as where , , , , and denotes the power of noise.
Estimation routine
Synthesizing the previous description of proposed algorithm, the proposed algorithm can be performed by the procedure in Table 1.
Complexity analysis
In this section, we compare the complexity of different PM algorithms. To be fair, we suppose that the used multiple-parallel linear arrays for all PM algorithms consist of L sensors. Since the number of sensors for PM algorithms [11], [12], [14] is odd, we let L be an odd. Suppose that a -element array and a -element array is used for the algorithms [11], [12], [14]. Assume that , so the computation mainly focuses on the construction of covariance matrix. The complexity
Simulation experiment
In this section, we give some simulation results to prove the effectiveness of the extended 2D PM algorithm based on the proposed three-parallel linear array. We compare the proposed method with the homologous 2D PM algorithms [11], [12], [13], [14], DOA matrix algorithm [7] and polynomial root-finding algorithm [9]. Just because of the large distance of any pair of sensors, the mutual coupling effect is minimal based on the proposed array configuration. Hence the proposed array has the obvious
Conclusion
In this paper, a three-parallel array with large element-spacing is proposed to eliminate the effect of mutual coupling. Based on this array, a high-precision 2D PM algorithm also is proposed, by which unambiguous and automatic match 2D DOA estimation can be got. Simulation results show the RMSE of the proposed method is far less than that of many existing 2D PM algorithms based on multiple-parallel linear array. Simulation results also can prove that the estimation accuracy can't descend with
CRediT authorship contribution statement
Sheng Liu: Conceptualization, Writing Original draft preparation.
Jing Zhao: Designing Experiments.
Decheng Wu: Writing-Reviewing and Editing.
Hailin Cao: Methodology.
Zhi Mao: Analyzing-Results.
Yiwang Haung: Performing-Experiments.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Sheng Liu received Ph.D. degree in school of information and communication engineering from Chongqing University in 2016. Now he is a professor in Tongren University. His research is focused on array signal processing and wireless sensor network.
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Cited by (4)
Joint Estimation of DOA and Mutual Coupling via Imposing an Annihilation Constraint
2022, Digital Signal Processing: A Review JournalCitation Excerpt :Among the various array imperfections, the mutual coupling arises from wave propagation and thus is hard to avoid. Therefore, mutual coupling calibration receives much attention [5–8]. As we know, the online calibration, which calibrates the array while estimating the DOAs, is more advantageous than the offline calibration [9].
2D DOA estimation by a large-space T-shaped array
2022, Digital Signal Processing: A Review JournalCitation Excerpt :Uniform circular array [1], parallel array [2–5] and L-shaped array [6–13] are three common planar arrays used for 2D DOA estimation. Parallel array is composed of multiple parallel linear arrays, and it has many array forms including two-parallel array [2], [3], three-parallel array [4], multiple-parallel array [5]. The L-shaped array consists of two linear arrays perpendicular to each other, which has better performance in 2D DOA estimation than parallel array [7].
Sheng Liu received Ph.D. degree in school of information and communication engineering from Chongqing University in 2016. Now he is a professor in Tongren University. His research is focused on array signal processing and wireless sensor network.
Jing Zhao received master degree from Xi'an University of Science and Technology in 2011. Now she is associate professor in Tongren university. Her research is focused on statistical signal processing.
Decheng Wu received Ph.D. degrees in circuits and systems from Chongqing University, Chongqing, China, in 2020. Now he is a lecturer with the School of Automation, Chongqing University of Posts and Telecommunications. His research is focused on deep learning and signal processing.
HaiLin Cao received the B.E. degree in electronic engineering and the M.S. and Ph.D. degrees in circuits and systems from Chongqing University, Chongqing, China, in 2002, 2006, and 2010, respectively. He is currently a Professor with the Chongqing Key Laboratory of Space Information Network and Intelligent Information Fusion, Chongqing University. His research interests include sensor networks, large structural health monitoring technology, and antenna and array signal processing.
Zhi Mao received Ph.D. degree from the school of mathematics and computational science, Xiangtan University, China, in 2015. He is currently professor in Tongren university. His research is focused on numerical solution of fractional differential equation and fractional order optimal control system.
Yiwang Huang received Ph.D. degree from the school of computer, Wuhan University, China, in 2015. He is currently professor in Tongren University. His research is focused on artificial intelligence, business process management and formal method.
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This work was supported by the Foundation of Top-notch Talents by Education Department of Guizhou Province of China (KY [2018]075), the Natural Science Foundation of Chongqing (cstc2021jcyj-bsh0198), the National Natural Science Foundation of China (U831117, 61763004, 62066040, 51877015), the Science and Technology Foundation of Guizhou Province of China ([2018]1162) and PhD Research Start-up Foundation of Tongren University (trxyDH1710).