**1. Introduction**

Direction-of-arrival (DOA) estimation of electromagnetic (EM) signals has attracted wide attention in many communication fields, such as radar [1,2], mobile networks [3] and sonar [4]. It is clear that DOA estimation is the basic and essential part in an array signal processing system. For example, a corresponding transmitting or receiving beamformer can be designed to extract signals in the direction of interest and suppress uninteresting interference signals. The electromagnetic vector sensor (EMVS) can catch polarization-related information compared to a conventional scalar sensor, which can further improve the target resolution, anti-interference ability and detection stability for DOA estimation [5–7]; therefore, the research for EMVS array direction finding has become a hotspot.

With the appearance of the Long-Vector MODEL (LV-MODEL) [5] (built for EMVS), multiple researchers have proposed various DOA estimators. The existing estimators can be summarized into three categories: (1) research on DOA estimators transplanting from scalar sensor; (2) research based on special array arrangement; (3) research based on advanced mathematical tools.

In terms of transplantation, the classic subspace-based super-resolution algorithm [8] (Multiple Signal Classification—MUSIC) was transplanted to the EMVS [9–11] array, but the algorithms often suffer high computational complexity because of the four-dimensional parameter search for two direction angles and two additional polarization angles; therefore, Weiss [12] used the polynomial root to reduce the computational complexity to a certain extent. In addition, another subspace-based super-resolution algorithm [13,14] (Estimation of Signal Parameters via Rotational Invariance Techniques—ESPRIT) was also transplanted

**Citation:** Wang, R.; Wang, Y.; Li, Y.; Cao, W.; Yan, Y. Geometric Algebra-Based ESPRIT Algorithm for DOA Estimation. *Sensors* **2021**, *21*, 5933. https://doi.org/10.3390/ s21175933

Academic Editors: Zihuai Lin and Wei Xiang

Received: 28 July 2021 Accepted: 27 August 2021 Published: 3 September 2021

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into the EMVS array, and realized closed-form estimation of DOA. In [15,16], authors showed that the statistical performance of the maximum likelihood and subspace-fitting algorithms based on the EMVS array are better than both MUSIC and ESPRIT, but the high calculation limits its application in actual engineering.

There are few studies based on the special array arrangemen<sup>t</sup> because most EMVS arrays are co-centered, leading to the mutual coupling interference and spatial information loss. In [17], a double-parallel-line EMVS array whose six components are all spatially separated achieved mutual coupling reduction to refine the DOA-finding accuracy by orders of magnitude. A triangular array [18] combined with a vector cross product and interferometric angle measurement, aimed to overcome the drawback that [17] cannot achieve two-dimensional aperture expansion. In addition, a spatial expansion method of a triangle structure [19] was proposed to provide higher-precision DOA estimation.

The traditional model for EMVS is just a linear combination of each component, which somehow locally destroy the orthogonality of the signal components [20]. Meanwhile, the heavy computational efforts and memory requirements during data processing for the DOA estimation cannot be ignored [21]. Recently, the hypercomplex has been widely studied and applied in multi-dimensional parameter estimation. Miron et al. [22] first proposed a new Quaternion Model (Q-MODEL) for the two-component EMVS array. Then, many models and algorithms based on quaternion have been proposed [23–25]; however, the Q-MODEL had to discard some of the original information because the quaternion only has three imaginary parts. Further, the research has extended to biquaternion [26] and quad-quaternion [27,28]. These quaternion-based algorithms showed higher estimation accuracy and less complexity; however, Jiang et al. [21] found that the physical interpretations of the presented quaternion-like models have not been discussed. In order to solve the problem, they derived G-MODEL [21] by Geometric Algebra (GA) formulations of Maxwell equations. The computing technology of G-MODEL not only minimizes the memory requirements and computational complexity, but also removes the correlation of noise on different antennas.

It is easy to find that the current studies utilizing hypercomplex algebra are mainly focused on the MUSIC algorithm [22,26–28]. In fact, MUSIC greatly suffers from a heavy computational burden for its spectrum search, while the computation of ESPRIT algorithm is cheaper, and it can automatically decouple [29]; therefore, the research in this paper extends the ESPRIT algorithm using a new mathematical tool—GA. Through the new calculation rules, the physical nature of EMVS is matched with the signal processing technology, which avoids correlation loss between different components in the previous algorithms. The major contributions of this paper are as follows.


The rest of this paper is organized as follows. Section 2 introduces the basics of GA and the EMVS model for narrow-band signals based on GA. Section 3 describes the proposed GA-ESPRIT in detail. Experimental results and analysis are provided in Section 4, followed by concluding remarks in Section 5.
