International Journal of Antennas and Propagation

Volume 2017 (2017), Article ID 9489318, 6 pages

https://doi.org/10.1155/2017/9489318

## Accurate 2-D AOA Estimation and Ambiguity Resolution for a Single Source under Fixed Uniform Circular Arrays

Department of Microwave Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China

Correspondence should be addressed to Le Zuo

Received 25 July 2017; Revised 7 October 2017; Accepted 22 October 2017; Published 12 December 2017

Academic Editor: Yuan Yao

Copyright © 2017 Le Zuo and Jin Pan. 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

This paper presents an analytic algorithm for accurate two-dimensional (2-D) angle of arrival (AOA) estimation of a single source under fixed uniform circular arrays (UCAs). Algebraic and explicit formulations for 2-D AOA estimation are first developed in the Fourier domain. It is shown that three is the minimum number of antennas for 2-D AOA estimation based on phase measurement. Then a signal model for phase extraction is established with equivalent phase noises through observations of signal samples corrupted by additive Gaussian white noise. Under fixed UCAs, 2-D AOA estimation of a single source would suffer from phase ambiguity, and hence, ambiguity resolution is also addressed in the Fourier domain by integer search. Numerical examples are provided to verify the effectiveness and appealing performance of the proposed 2-D AOA estimation algorithm.

#### 1. Introduction

Fast and accurate estimation of the two-dimensional (2-D) angle of arrival (AOA) of incident plane waves is important in array signal processing due to its applications in radar, sonar, and mobile communications. Uniform circular array (UCA) is extensively utilized in the context of 2-D AOA estimation due to its attractive advantages, including 360° azimuthal coverage, almost unchanged directional pattern, and additional elevation angle information [1, 2]. Recently, a lot of literatures have been reported on the AOA estimation with UCAs. In [3], spatial averaging algorithm and spectrum search were applied to 2-D estimation with UCAs. Reference [4] developed two eigenstructure-based algorithms for 2-D estimation with UCAs. However, these methods introduce errors in the form of bias and excess variance, and hence, the obtained estimates may be far from optimal [5]. Furthermore, these methods involve eigenvalue decomposition; thus, the computational load is significant. Liao et al. proposed a generalized algorithm for 2-D AOA estimation based on the least square estimation [2]. As will be shown in Section 2, if the antenna element number is even, our algorithm is equivalent to the generalized method. However, if the antenna element number is odd, the simulated accuracy by the generalized method is lower than our algorithm, as will be shown in Section 4.

In addition, it is well known that high AOA estimation accuracy can be obtained from large apertures. However, the measurement of phase difference can only be made modulo of , which leads to an ambiguity in determining the AOA of the source [6]. To solve the phase ambiguity, a modulo conversion method [7] was proposed, but it is inherently developed for linear array interferometers and cannot be directly applied to UCAs, for UCA’s phase differences are dependent on both elevation and azimuth angles. As regards ambiguity resolution under UCAs, rotary ways were used [8–10], whereas rotary interferometers face the problem of source correspondence and real-time applications. In [11], a method called subarray grouping and ambiguity searching was proposed and the rough angle estimation was achieved by searching the nearest value among subarrays. However, the antenna elements were in pairs, and hence, the number of antenna elements must be even.

To avoid eigenvalue calculation, in this paper, we propose an analytical 2-D AOA estimation algorithm under fixed UCAs. The algorithm is based on the Fourier analysis of the phase around the circular aperture. The underlying AOA estimation problem is reformulated as expansion coefficient calculation problem. The solutions to 2-D AOAs are explicit discrete Fourier transform (DFT) of antenna outputs that sample the phases around the circular aperture. A signal model for phase extraction is then established with equivalent phase noises through observations of signal samples corrupted by additive white Gaussian noise (AWGN). Furthermore, without rotation, we address ambiguity resolution by finding the missing spectrum of ambiguity numbers through integer search. Numerical examples show the effectiveness and appealing performance of the proposed algorithm.

This paper contributes to the area of 2-D AOA estimation in the following aspects: (1)Algebraic formulations for accurate 2-D AOA estimation under UCAs are presented with low computational complexity.(2)The estimation algorithm sufficiently exploits the centrosymmetry and periodicity of the circular aperture by Fourier transform, resulting in an algebraic solution to 2-D AOAs.(3)A novel ambiguity resolution based on integer search and inverse Fourier transform is developed for fixed UCAs, and hence, it is applicable to real-time AOA estimations.

The rest of this paper is organized as follows. In Section 2, phase-based expressions for continuous and discrete phases around a circular aperture are first developed, which decouples the 2-D AOA parameters by Fourier transform, and then a signal model is established for phase extraction in AWGN. Section 3 addresses ambiguity resolution based on DFT and integer search. Numerical simulations are presented in Section 4. Section 5 concludes this paper.

#### 2. AOA Estimation Algorithm

In this section, Fourier transform is first applied to the noiseless periodic phase distribution around a continuous circular aperture and then to the noiseless discrete phase samples. In order to extract the phases for the AOA estimation, a signal model is established, in which the phase noises are equivalent to the AWGN in the time domain. Moreover, the proposed algorithm is compared to a previous method [2].

##### 2.1. Continuous Aperture

Consider a circular aperture located at in the spherical coordinate system of , as shown in Figure 1. The phase of the electric field of an incident wave from to can be written as
where the azimuth angle is measured counterclockwise from the *x*-axis and the elevation angle is measured down from the *z*-axis, is the wave number in free space, and is the wavelength. The first term depends on the element position and contains the unknown AOA parameters. The second term, that is, , is a constant and represents the initial phase of the incident wave, which can be interpreted as the phase of the incident wave arriving at the center of the array.