International Journal of Antennas and Propagation

Volume 2015 (2015), Article ID 485351, 10 pages

http://dx.doi.org/10.1155/2015/485351

## Two-Dimensional DOA Estimation for Uniform Rectangular Array Using Reduced-Dimension Propagator Method

^{1}Key Laboratory of Radar Imaging and Microwave Photonics (Nanjing University of Aeronautics and Astronautics), Ministry of Education, Nanjing 210016, China^{2}Institute of Command Information System, PLA University of Science and Technology, Nanjing 210007, China

Received 17 May 2014; Revised 3 August 2014; Accepted 17 August 2014

Academic Editor: Ahmed Shaharyar Khwaja

Copyright © 2015 Ming Zhou 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

A novel algorithm is proposed for two-dimensional direction of arrival (2D-DOA) estimation with uniform rectangular array using reduced-dimension propagator method (RD-PM). The proposed algorithm requires no eigenvalue decomposition of the covariance matrix of the receive data and simplifies two-dimensional global searching in two-dimensional PM (2D-PM) to one-dimensional local searching. The complexity of the proposed algorithm is much lower than that of 2D-PM. The angle estimation performance of the proposed algorithm is better than that of estimation of signal parameters via rotational invariance techniques (ESPRIT) algorithm and conventional PM algorithms, also very close to 2D-PM. The angle estimation error and Cramér-Rao bound (CRB) are derived in this paper. Furthermore, the proposed algorithm can achieve automatically paired 2D-DOA estimation. The simulation results verify the effectiveness of the algorithm.

#### 1. Introduction

Direction-of-arrival (DOA) estimation is a fundamental problem in array signal processing and has been widely used in many fields [1–5]. Till now, many algorithms have been proposed for DOA estimation with uniform linear array. Among them, multiple signal classification (MUSIC) algorithm [6] and estimation of signal parameters via rotational invariance technique (ESPRIT) algorithm [7] are widely used superresolution methods. The two-dimensional DOA (2D-DOA) estimation problem, which plays an important role in array signal processing, has received more attention. This problem is usually considered with rectangular array, two parallel uniform linear arrays, L-shape array, and so forth. Also, many algorithms have been considered for 2D-DOA estimation, which include 2D-MUSIC algorithm [8], 2D Unitary ESPRIT algorithm [9], modified 2D-ESPRIT algorithm [10], matrix pencil methods [11, 12], maximum likelihood method [13], parallel factor (PARAFAC) algorithm [14], and high order cumulant method [15].

Propagator method, which is known as a low complexity method without eigenvalue decomposition (EVD) of the covariance matrix of the received data, has been proposed for DOA estimation through peak searching [17–20]. Due to its low complexity, PM algorithms are widely used for 2D-DOA estimation. In [21], a PM-based DOA estimation algorithm is proposed for two parallel uniform linear arrays via the rotational invariance property of propagator matrix, which requires extra pairing match. Reference [22] presents an efficient 2D-DOA estimation algorithm with two parallel uniform linear arrays using PM. In [23], an improved PM algorithm is proposed for 2D-DOA estimation, which has better angle estimation performance than the algorithms in [21, 22]. The above-mentioned PM algorithms can be extended to the rectangular array for 2D-DOA estimation; however, they only employ the rotational invariance property of propagator matrices, and as a result of that, the accuracy of angle estimation performance is not sufficient in some cases, especially with low signal to noise ratio (SNR). Two-dimensional PM (2D-PM) algorithm through peak searching can be extended for 2D-DOA estimation; however, the high computational complexity caused by 2D peak searching makes it inefficient.

In this paper, we derive a reduced-dimension PM (RD-PM) algorithm, which reduces the high complexity for 2D-DOA estimation with uniform rectangular array compared with 2D-PM algorithm. The proposed algorithm applies the rotational invariance property of propagator matrix to get the initial angle estimation and then employs one-dimensional local searching to get more accurate angle and finally obtains the other angle via least square (LS) method and estimate azimuth and elevation angles. The proposed algorithm has the following advantages: the proposed algorithm has lower computational complexity than 2D-PM algorithm since it requires no EVD of the covariance matrix of the receive data and it only requires one-dimensional local searching; the angle estimation performance of the proposed algorithm is better than that of ESPRIT algorithm and the PM algorithms in [21, 23], also very close to that of 2D-PM algorithm; it can obtain automatically paired parameters estimation.

The remainder of this paper is structured as follows: Section 2 shows the data model for uniform rectangular array, while Section 3 proposes the RD-PM algorithm for 2D-DOA estimation. The angle estimation error is derived as well as CRB in Section 4. In Section 5, the simulation results verify the feasibility and effectiveness of the proposed algorithm, and the conclusions are showed in Section 6.

*Notion*. , , , and denote transpose, conjugate-transpose, inverse, and pseudoinverse operations, respectively; stands for diagonal matrix whose diagonal element is a vector ; , , and denote a vector of ones, a identity matrix, and a vector of zeros, respectively. , , and are the Kronecker product, the Khatri-Rao product, and the Hadamard product, respectively; is to get the phase angle.

#### 2. Data Model

As illustrated in Figure 1, consider a uniform rectangular array having sensors; and are the numbers of sensors in -axis and -axis, respectively. The distance between the two adjacent elements is in both -axis and -axis. Assume that there are uncorrelated sources, and are the elevation and azimuth angles of the th source.