Reference sensor relocation-based thinned coprime array design for DOA estimation
Introduction
Direction of Arrival (DOA) estimation is an important topic of array signal processing, which is widely used in radar, sonar, navigation, wireless communication, ultrasonic imaging, and other fields [1], [2], [3]. As the most commonly used array configuration, the uniform linear array (ULA) has the most extensive applications. However, there are also several issues for ULA which are as follows. Firstly, it requires that the inter-element spacing should be kept fixed and not exceed half wavelength, which easily leads to a high cost and is notoriously difficult to be equipped in the special geomorphological environment. Secondly, the dense inter-element spacing makes the array susceptible to the mutual coupling effect [4], [5], [6], which affects the accuracy of the DOA estimator. Furthermore, the traditional algorithms such as Multiple Signal Classification (MUSIC) [7], Maximum Likelihood (ML) [8], and Estimation of Signal Parameters via Rotational Invariable Technique (ESPRIT) [9], [10] suffer from a low identification capability of the number of sources.
To address these problems, sparse array designs, such as minimum redundant arrays (MRAs) [11], the minimum hole arrays (MHAs) [12], nested arrays (NAs) [13], [14], [15], [16], [17], [18], and coprime arrays (CAs) [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], have been suggested one after another. MRA is designed by maximizing the number of consecutive virtual sensors, and similar to MRA, MHA is constructed by minimizing the number of holes. However, MRA and MHA do not get closed-form expressions related to the sensor locations and the achievable degrees of freedom (DOFs), and thus it is difficult to obtain a proper array configure when the number of sensors is very large. Different from the above arrays, NA [13] has a clear expression related to the inter-element spacing and the number of sensors, and its hole-free property in the difference coarray is also attractive enough for subspace-based methods. Moreover, NA can achieve DOFs with physical sensors, and its mutual coupling effect between sensors is weaker than the case in ULA. To further enhance DOA estimation, an improved nested array is proposed in the literature [14] by increasing the inter-element spacing and adding one sensor. However, it is still vulnerable to the mutual coupling effect since the inter-element spacing is less than or equal to half wavelength in one of the two subarrays. To overcome this drawback, in the augmented nested array (ANA) [15], the dense uniform linear array is divided into two or four parts while keeping the number of array sensors constant, and the explicit expression of the sensor positions is derived. Compared with NA, ANA has higher DOFs and effectively alleviates the mutual coupling. Towards this end, the maximum inter-element spacing constraint-based array [16] is composed of three ULA and two separate sensors, providing a large number of DOFs and a low mutual coupling effect.
Similar to NA, CA [19] is another hot sparse array configuration that consists of two uniform linear subarrays, in which, assuming where M and N is a pair of coprime integers, one subarray has M sensors with the inter-element spacing being N times half-wavelength, while the other subarray has N sensors with the inter-element spacing setting to M times half-wavelength. Resultantly, by using CA, we could identify MN uncorrelated signals with sensors and alleviate mutual coupling. As there are limited consecutive lags and exist large amounts of holes in the difference coarray generated by CA, the augmented CA (ACA) is proposed in the literature [20] which extended one of the subarrays to 2M, lengthening the virtual array aperture. To reduce the redundancy of sensors in ACA, a thinned coprime array (TCA) [21] is offered by removing parts of sensors that would not affect the DOFs of ACA. In [22], [23], the coprime arrays with compressed inter-element spacing (CACIS) and displaced subarrays (CADiS) are proposed to increase consecutive and unique lags of difference coarray. However, there are still some holes, and the number of consecutive lags is not enough in the above two methods so that all the lags generated by the difference coarray cannot be used effectively for the traditional algorithm (such as MUSIC, ESPRIT).
Existing solutions to the hole problem are mainly classified into three categories: the first is that matrix completion is used to recover missing information in the hole position [24], [25], [26]. The second goes to convert the DOA estimation problem into a sparse signal recovery problem by using the compressed sensing theory [27]. Both types of algorithms suffer from high computational complexity. The third type is to fill the holes by exploiting array motions. For example, in [28] the array is displaced forward a half wavelength, and the received array vectors before and after the move are jointly processed to perform DOA estimation, filling the hole positions. By moving one subarray of ACA to set location and another subarray is flipped around the reference sensor, sliding extended coprime array (SECA) is formed for DOA estimation, while relocating extended coprime array (RECA) takes some sensors of a subarray of CADIS to a specific location, increasing the number of consecutive lags and reducing the redundancy of array sensors [29].
Since moving the whole array is impracticable in some scenarios, we focus on the motion of specific sensors to construct a novel sparse array, decreasing the number of holes and increasing the maximum number of consecutive lags. The main contributions of this paper are summarized as the following:
(1) From the perspective of hole positions, the closed-form expressions about the range of consecutive lags and the number of physical sensors are proved for TCA.
(2) By analyzing the position relationship between consecutive lags and the given number of sensors, the 0-position sensor is moved to a set position, developing a novel sparse array.
(3) The achievable number of consecutive lags and unique lags of the proposed array are derived and compared with the existing sparse arrays.
The remainder of this paper is as follows. The signal model of TCA is described in Section 2. Several propositions related to the 0-position sensor are analyzed on consecutive lags and unique lags of TCA, and the proposed sparse array and its theoretical analysis are also provided in Section 3. Performance evaluation of the proposed array is demonstrated through simulation results in Section 4. Conclusions are drawn in Section 5.
Throughout this paper, we use lower-case (upper-case) boldface characters to denote vectors (matrices), respectively. denotes the largest integer less than or equal to a. The superscripts , and respectively denote transpose, the conjugate and conjugate transpose of a vector or matrix. ⊗ and ⊙ stand for the Kronecker product and the Khatri-Rao product, respectively. In addition, generates a diagonal matrix whose diagonal elements are the elements of a and denotes the vectorization operator. denotes the expectation operator. means the complexity. represents the identity matrix.
Section snippets
Signal model
TCA [21] is composed of three ULAs where one subarray has N sensors with inter-element spacing being Md, and the other two subarrays have respective and sensors with inter-element spacing being Nd, where d is the unit of inter-element spacing, typically set to half wavelength, i.e., . Without loss of generality, we assume that M and N are a pair of coprime integers and require , meanwhile, the first sensor of two of three subarrays coincides and is denoted as the reference
Proposed array structure
By moving the 0-position sensor of TCA to the position , , a novel array is proposed in this section, which is shown in Fig. 2. The position set of the physical sensors in the proposed array is denoted by , which satisfies the following relationship: where
It is noted that the 0-position sensor of the subarrays is moved to the set location together, i.e., the reference sensor of the
Sparse array analysis
In this section, to demonstrate the structure formation and performance indicators of the proposed array, we will analyze TCA (before moving), the sensor to be moved (reference sensor), and the proposed sparse array (after moving) in terms of hole positions, consecutive lags and unique lags in the corresponding difference coarrays. The core idea of our work is that by using the derived mathematical expression of the hole positions of the difference coarray in TCA, a novel filling strategy for
Simulation results
In this section, numerical simulations are conducted to illustrate the performance of the proposed array compared with NA [13], ACA [20], TCA [21], SECA [29], RECA [29] for DOA estimation. To make a fair comparison, the same number of physical sensors and the same DOA estimation algorithms (MUSIC and ESPRIT [30]) are chosen as the preconditions. Array configurations are as follows. For NA, the array parameters are set to , . In TCA and proposed array, coprime integers are set to and
Conclusion
Motivated by TCA, a novel sparse array is constructed in this paper, by relocating the 0-position sensor to the set position. Resultantly, we get a sparse array configuration with increased consecutive lags, enhanced DOFs, reduced mutual coupling and decreased sensor redundancy. Particularly, its closed-form expression related to consecutive lags and unique lags of difference coarray is derived, according to the properties of hole positions. In the simulation part, the performance of DOA
CRediT authorship contribution statement
Ke Liu: Methodology. Zezheng Zhu: Investigation, Writing – original draft. Junda Ma: Writing – review & editing.
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.
Ke Liu received his B.S. degree in 2010 from Harbin University of Science and Technology and his M.S. and Ph.D. degree in 2013 and 2018 from Harbin Engineering University. He is currently a faculty in Harbin University of Science and Technology. His research interests include array signal processing, compressive sensing and adaptive filter.
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Cited by (1)
Ke Liu received his B.S. degree in 2010 from Harbin University of Science and Technology and his M.S. and Ph.D. degree in 2013 and 2018 from Harbin Engineering University. He is currently a faculty in Harbin University of Science and Technology. His research interests include array signal processing, compressive sensing and adaptive filter.
Zezheng Zhu received B.S. degree in communication engineering from Mudanjiang Normal University in 2019. He is currently pursuing his M.S. degree in Harbin University of Science and Technology. His research interests include DOA estimation and array design.
Junda Ma received his M.S. and Ph.D. degree in 2013 and 2018 in Harbin Engineering University, respectively. Now, he is a faculty with the School of Automation in Harbin University of Science and Technology. His research interests include signal processing and control algorithm.