International Journal of Antennas and Propagation

Volume 2016 (2016), Article ID 7832475, 9 pages

http://dx.doi.org/10.1155/2016/7832475

## DOA Estimation of Cylindrical Conformal Array Based on Geometric Algebra

Department of Electronic Science and Engineering, National University of Defense Technology, Deya Road 109, Changsha 410073, China

Received 7 July 2016; Revised 18 October 2016; Accepted 2 November 2016

Academic Editor: Youssef Nasser

Copyright © 2016 Minjie Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Due to the variable curvature of the conformal carrier, the pattern of each element has a different direction. The traditional method of analyzing the conformal array is to use the Euler rotation angle and its matrix representation. However, it is computationally demanding especially for irregular array structures. In this paper, we present a novel algorithm by combining the geometric algebra with Multiple Signal Classification (MUSIC), termed as GA-MUSIC, to solve the direction of arrival (DOA) for cylindrical conformal array. And on this basis, we derive the pattern and array manifold. Compared with the existing algorithms, our proposed one avoids the cumbersome matrix transformations and largely decreases the computational complexity. The simulation results verify the effectiveness of the proposed method.

#### 1. Introduction

A conformal antenna is an antenna that conforms to a prescribed shape. The shape can be some part of an airplane, high-speed missile, or other vehicle [1]. Their benefits include reduction of aerodynamic drag, wide angle coverage, and space-saving [2]. Nevertheless, due to the complex curved surface structure, the pattern of each antenna is inconsistent. Thus, the conformal array can no longer be regarded as a simple isotropic one. The pattern multiplication theorem is not available as well. Most classical DOA estimation algorithms cannot be directly transplanted to such scene.

In recent years, there has been a considerable interest in estimating DOAs for conformal array. Milligan used Euler rotation angles to find the patterns with elements in a conformal array that requires one to rotate not only the direction but also the polarization [3]. In [2], Wang et al. proposed a uniform method for the element polarized pattern transformation of arbitrary 3D conformal arrays based on Euler rotation. Yang et al. introduced a conformal array DOA algorithm with an unknown source number; the method was realized by virtue of the pseudo expected signal [4]. However, the root mean square error (RMSE) deteriorated severely when the number of snapshots was small. Up to the present, we have observed that most of DOA estimation algorithms for conformal array are based on the Euler rotation transformation which converts the local coordinate system to the global coordinate system. Though the Euler rotation angle is a useful tool for spatial rotation transformation [5], a huge amount of computation is incurred.

Geometric algebra is the largest possible associative algebra that integrates all algebraic systems (algebra of complex numbers, matrix algebra, quaternion algebra, etc.) into a coherent mathematical language [6]. Three-dimensional pattern analysis of arbitrary conformal arrays using the mathematical framework of the geometric algebra was introduced [7]. Nevertheless, this mathematical language was not transplanted to the DOA estimation. In [8], Zou et al. took several elements as a new one and transformed the original array into another regular array to estimate the DOA. However, this method was only suitable for some particular array structures. Combining the MUSIC with geometric algebra to solve the DOA estimation has not been addressed in the literature. In this paper, we fill this gap and study the problem based on the cylindrical conformal array. Compared with the existing methods, the proposed one has three main advantages. Firstly, it does not need to calculate the rotation matrices and therefore has a much lower computational complexity. Subsequently, it is not limited to the cylindrical conformal array due to its strong commonality. Finally, it can still work effectively even when the number of polarized signals is larger than that of the array elements.

The structure of this paper is as follows. In Section 2, the rotors in geometric algebra which establishes the mathematical knowledge of transformation is briefly introduced. In Section 3, we derive the cylindrical conformal array manifold using rotors and present the GA-MUSIC algorithm. In addition, to better explain the superiority of GA-MUSIC in reducing the computational complexity, we briefly introduce the Euler angle and compare it with the proposed algorithm. Simulations using the proposed method for cylindrical conformal array are given in Section 4. Finally, the conclusions are drawn.

#### 2. Rotors in Geometric Algebra

Geometric algebra was first introduced by the British mathematician, named Clifford, in the nineteenth century. He constructed the geometric product by combining the inner product with the outer product. The main advantage of the geometric algebra is embodied in processing the rotation transformation [9]. Various rotations can be described by an element called the rotor. A rotor is more general than an Euler rotation angle because a rotor can be used in an arbitrary dimensional space.

We begin by introducing a new product between vectors that we call the outer product. Let us use the wedge symbol “” to denote outer product with the properties listed in Table 1.