International Journal of Antennas and Propagation

Volume 2015, Article ID 323545, 10 pages

http://dx.doi.org/10.1155/2015/323545

## A Low Complexity Subspace-Based DOA Estimation Algorithm with Uniform Linear Array Correlation Matrix Subsampling

School of Electronic and Electrical Engineering, Hongik University, Mapo-gu, Wausan-ro 94, Seoul 04066, Republic of Korea

Received 9 July 2015; Revised 8 November 2015; Accepted 12 November 2015

Academic Editor: Ding-Bing Lin

Copyright © 2015 Do-Sik Yoo. 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

We propose a low complexity subspace-based direction-of-arrival (DOA) estimation algorithm employing a direct signal space construction method (DSPCM) by subsampling the autocorrelation matrix of a uniform linear array (ULA). Three major contributions of this paper are as follows. First of all, we introduce the method of autocorrelation matrix subsampling which enables us to employ a low complexity algorithm based on a ULA without computationally complex eigenvalue decomposition or singular-value decomposition. Secondly, we introduce a signal vector separation method to improve the distinguishability among signal vectors, which can greatly improve the performance, particularly, in low signal-to-noise ratio (SNR) regime. Thirdly, we provide a root finding (RF) method in addition to a spectral search (SS) method as the angle finding scheme. Through simulations, we illustrate that the performance of the proposed scheme is reasonably close to computationally much more expensive MUSIC- (MUltiple SIgnal Classification-) based algorithms. Finally, we illustrate that the computational complexity of the proposed scheme is reduced, in comparison with those of MUSIC-based schemes, by a factor of , where is the number of sources and is the number of antenna elements.

#### 1. Introduction

Subspace-based spectral estimation and direction-of-arrival (DOA) estimation schemes have been widely studied during the last several decades [1–3]. Right after the introduction of Pisarenko and MUSIC (MUltiple SIgnal Classification) algorithms, various alternative schemes such as min-Norm method [4], ESPRIT (Estimation of Signal Parameter via Rotational Invariance Technique) [5], and root-MUSIC method [6] were developed during the first decade. In more recent years, such early ideas and concepts have been extended to deal with various issues such as nonlinear or nonuniform array shapes [7–9], coherent sources [10], and 2-dimensional (2D) angle estimation [11]. Among such issues drawing continued research interest is the computational complexity, which is particularly important for arrays with large number of antenna elements.

One of the main reasons for large computational complexity with existing subspace-based algorithms is the computational burden to construct the subspace, which usually involves the process of eigendecomposition (ED) or singular-value decomposition (SVD). To reduce the computational burden that arises due to ED or SVD, various algorithms that do not require ED or SVD have been proposed. Among the first such examples was the method proposed in [12] which computes the projection matrix by using a submatrix of the correlation matrix. Later, in [13], an improved algorithm called orthogonal propagator method has been proposed that achieves better performance particularly at medium and high signal-to-noise ratio (SNR). Recently, a 2-dimensional (2D) DOA estimation algorithm based on cross-correlation matrix was proposed in [14] that similarly computes the required projection operators without using ED or SVD. However, even though these methods do not require ED or SVD process, they are still computationally burdensome particularly for an array with large antenna elements.

More recently, Xi and Liping proposed, in [15], a very impressive 2D DOA estimation algorithm with L-shaped array. Most of all, it is truly computationally efficient reducing the computational complexity by a factor of , additionally from the algorithms in [12–14]. The authors also introduced a method to exploit the conjugate symmetry to enlarge the effective array aperture, which resulted in reasonably impressive performance. However, despite the fact that the performance of the algorithm [15] was reported to outperform that of [14], its performance is still inferior to computationally more expensive MUSIC scheme particularly at low signal-to-noise ratio (SNR). Moreover, based on the cross-correlation matrix between data collected by two component ULAs of L-shaped array, it cannot be used for a single ULA.

In this paper, we propose a low complexity subspace-based DOA estimation algorithm employing direct signal space construction method (DSPCM) similar to that in [15] but by subsampling the autocorrelation matrix of a ULA. The use of autocorrelation matrix in signal space construction enables us to use the low complexity method with a single ULA and to exploit the Toeplitz structure of the autocorrelation matrix in improving the accuracy of estimating signal vectors. We further propose a method of signal vector separation to improve the distinguishability of the signal vectors and hence to improve the performance particularly in the low SNR regime. Finally, we derive root finding (RF) method in addition to the spectral search (SS) method as the angle finding scheme.

The rest of this paper is organized as follows. In Section 2, the system model is described together with the definition of the notations used throughout this paper. In Section 3, the theoretical background of the algorithm is provided. In particular, a sufficient set of conditions are provided for the applicability of the algorithm. Next, the issues of practical algorithm implementation are discussed in Section 4. In this section, we provide two realizations, namely, SS and RF schemes of the proposed DSPCM. Then, the performance of the proposed scheme is compared with existing schemes, particularly with the MUSIC-based schemes and with that proposed in [15], and the computational complexity is analyzed in Section 5. Finally, we draw conclusions and discuss the directions of future work in Section 6.

*Notation.* The set of complex numbers will be denoted by and the set of all complex matrices will be denoted by . The complex conjugate, the transpose, and the conjugate transpose of matrix are denoted, respectively, by , and Boldface letters are used for matrices and vectors. We will denote by the matrix whose , and th columns are , and , respectively. identity matrix and zero matrix are denoted, respectively, by and We will denote by the Dirac delta function. For a real number , denotes the largest integer that does not exceed

#### 2. System Model

In this paper, we consider a uniform linear array (ULA) and propose a low complexity subspace-based direction-of-arrival (DOA) estimation algorithm based on correlation matrix subsampling. We assume that antenna elements are located along the -axis at coordinates as shown in Figure 1. We assume that narrowband far-field plane waves of wavelength impinge on the array making angles with the -axis. We let denote the signals corresponding to these waves arriving at the origin, namely, at the leftmost antenna element in Figure 1. We denote by the data collected by the antenna elements located at , respectively. For notational convenience, we let