International Journal of Antennas and Propagation

Volume 2015, Article ID 127621, 8 pages

http://dx.doi.org/10.1155/2015/127621

## Two-Dimensional Direction-of-Arrivals Estimation Based on One-Dimensional Search Using Rank Deficiency Principle

^{1}Harbin Institute of Technology at Weihai, Weihai 264209, China^{2}Harbin Institute of Technology, Harbin 150001, China

Received 21 September 2015; Accepted 14 December 2015

Academic Editor: Andy W. H. Khong

Copyright © 2015 Feng-Gang Yan 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 efficient method for two-dimensional (2D) direction-of-arrivals (DOAs) estimation is proposed to reduce the computational complexity of conventional 2D multiple signal classification (2D-MUSIC) algorithm with uniform rectangular arrays (URAs). By introducing two electrical DOAs, the formula of 2D-MUSIC is transformed into a new one-dimensional (1D) quadratic optimal problem. This 1D quadratic optimal problem is further proved equivalent to finding the conditions of noise subspace rank deficiency (NSRD), which can be solved by an efficient 1D spectral search, leading to a novel NSRD-MUSIC estimator accordingly. Unlike 2D-MUSIC with exhaustive 2D search, the proposed technique requires only an efficient 1D one. Compared with the estimation of signal parameter via rotation invariance techniques (ESPRIT), NSRD-MUSIC has a significantly improved accuracy. Moreover, the new algorithm requires no pair matching. Numerical simulations are conducted to verify the efficiency of the new estimator.

#### 1. Introduction

Estimation of direction-of-arrival (DOA) of multiple narrow-band signals using sensor arrays is of great interest in many fields such as radar, sonar, and wireless communications [1, 2]. Over several decades, this topic has been extensively addressed and numerical algorithms have been developed. Among those approaches, the multiple signal classification (MUSIC) [3] algorithm firstly exploits the orthogonality between the signal and the noise subspaces to achieve a so-called superresolution DOA estimate. Because those two subspaces are perfectly orthogonal to each other when the signal-to-noise ratio (SNR) is sufficiently high, the MUSIC algorithm can resolve two sources as closely spaced as possible in theory. Another outstanding advantage of the MUSIC algorithm over the other superresolution methods is that the former can be used with arbitrary array geometries [4]. However, the computational complexity of MUSIC is prohibitively expensive for real-time applications, owing to an involved matrix decomposition step for the signal or/and the noise subspace estimation and a tremendous spectral search step for final DOA estimates [5].

In practical applications, it is two-dimensional (2D) DOAs (2D DOAs) (i.e., elevation and azimuth angles) that people are usually most concerned about. Estimators such as MUSIC and estimation of signal parameter via rotation invariance technique (ESPRIT) [6] have been studied in one-dimensional (1D) situations and further extended to 2D scenarios [7]. Due to the increase in the dimensionality of the 2D DOA estimation problem, the complexity of DOA estimation process is severely affected by the array geometry [8], and further the pair matching (i.e., association or alignment) of the estimated elevation and azimuth angles is usually required [9]. Since, for arbitrary array configurations, the elevation and azimuth angles are extremely coupled with each other in the array response manifold, the heavy 2D spectral search cannot be avoided. To reduce the complexity, researchers are obliged to focus on reducing the computational burden of the 2D DOA estimation problem with various specified array structures.

Planar sensor arrays composed of two or more uniform linear arrays (ULAs) with simple geometry configurations have received considerable attention. Exploiting specified geometries such as L-shaped [10, 11], Y-shaped [12], and Z-shaped arrays [13], 2D DOAs of multiple incident signals can be estimated with reduced computational burden by applying most 1D subspace-based estimation methods including MUSIC and ESPRIT. Special array configurations allow utilizing the shift invariance and the partition of array response vector of two overlapping subarrays of each ULA such that the signal or/and noise subspaces can be estimated without eigenvalue decomposition (EVD) or singular valued decomposition (SVD). Nevertheless, spectral search is still required for those methods, and, as a matter of fact, the complexity of spectral search is substantially heavier than that of subspace decomposition, especially for 2D DOAs estimates [14]. Moreover, because the total arrays are usually divided into several subarrays in those methods, the array aperture is in fact reduced and DOA estimation accuracy may be sacrificed [15].

Uniform rectangular array (URA) is another frequently discussed array structure. A geometric formulation of the ESPRIT algorithm exploiting subarrays of URA is presented to reparameterize the weighted subspace fitting (WSF) algorithm in [16], leading naturally to an extension to the 2D DOA estimation problem. With high dimensional signal processing, the 2D subspace-based algorithms in [17] can provide precise estimates at the expense of high computational complexity exponentially increasing with the size of the rectangular arrays. To reduce the computational load, a tree structure algorithm by first performing 1D spatial smoothing and then 1D MUSIC several times successively is proposed in [18] to estimate the azimuth and elevation angles independently. Despite the computational efficiency, almost all of the aforementioned techniques realize 2D DOA estimates at the cost of a reduction in array aperture.

Recently, an outstanding reduced-dimensional MUSIC (RD-MUSIC) algorithm is proposed to avoid the high computational cost within 2D-MUSIC for direction of departure (DOD) and DOA estimation in multiinput multioutput (MIMO) radar [19]. Using DOA information embedded in the velocity sensors, RD-MUSIC starts 1D MUSIC searches for the DOD and DOA in succession without parameter pairing nor 2D search. In fact, the formula of RD-MUSIC is somewhat similar to that of 2D DOA estimates except that the noise projection matrix in the former is of reduced order and is invertible while that of the latter is not. Following this idea, we present in this paper a new efficient technique for 2D DOAs estimation with URAs. With two introduced electrical angles, we show that the formula of 2D-MUSIC can be transformed into a new 1D quadratic optimal problem. Further analysis demonstrates that this 1D quadratic optimal problem is equivalent to finding the conditions of noise subspace rank deficiency (NSRD), which can be solved by an efficient 1D spectral search. Consequently, a novel NSRD-MUSIC estimator at hand is derived. Unlike 2D-MUSIC with exhaustive 2D search, the proposed technique requires only an efficient 1D one. Compared with ESPRIT, NSRD-MUSIC has a significantly improved Root Mean Square Error (RMSE) performance.

The outline of this paper is as follows. The narrow-band signal model and the conventional 2D-MUSIC algorithm are introduced in Section 2. Two transformed electrical angles reformulating the 2D-MUSIC algorithm into a new 1D optimum problem is then considered, and the proposed method is addressed in detail in Section 3. The complexity of our method is analyzed in Section 4 and simulation results are conducted and discussed to validate the effectiveness of new method in Section 5.

#### 2. Signal Model and Standard 2D-MUSIC

Assume that there are narrow-band signals with unknown 2D DOAs simultaneously incident on URA of sensors, where and are the numbers of elements in -direction and -direction, respectively. The corresponding element spacing symbols are and , as depicted in Figure 1. The azimuth angle is defined as the one between the wave direction and the -axis while the elevation angle is defined as the one between the -axis and the projection of wave direction onto the plane. Note that and . It is assumed that there are snapshots available; array output vector at snapshot is given bywhereis the matrix of the signal direction vectors andis the steering vector. In addition, and is the vector of source waveforms; is the vector of white sensor noise; is the center wavelength; is the matrix transpose.