International Journal of Antennas and Propagation

Volume 2018, Article ID 9695326, 8 pages

https://doi.org/10.1155/2018/9695326

## Two-Step Root-MUSIC for Direction of Arrival Estimation without EVD/SVD Computation

Harbin Institute of Technology at Weihai, Weihai 264209, China

Correspondence should be addressed to Jun Wang; moc.liamg@oidutsgnawnhoj

Received 29 March 2018; Revised 27 June 2018; Accepted 11 July 2018; Published 6 August 2018

Academic Editor: Herve Aubert

Copyright © 2018 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

Most popular techniques for super-resolution direction of arrival (DOA) estimation rely on an eigen-decomposition (EVD) or a singular value decomposition (SVD) computation to determine the signal/noise subspace, which is computationally expensive for real-time applications. A two-step root multiple signal classification (TS-root-MUSIC) algorithm is proposed to avoid the complex EVD/SVD computation using a uniform linear array (ULA) based on a mild assumption that the number of signals is less than half that of sensors. The ULA is divided into two subarrays, and three noise-free cross-correlation matrices are constructed using data collected by the two subarrays. A low-complexity linear operation is derived to obtain a rough noise subspace for a first-step DOA estimate. The performance is further enhanced in the second step by using the first-step result to renew the previous estimated noise subspace with a slightly increased complexity. The new technique can provide close root mean square error (RMSE) performance to root-MUSIC with reduced computational burden, which are verified by numerical simulations.

#### 1. Introduction

Subspace-based algorithms for estimating the direction of arrivals (DOAs) of multiple signals impinging on array of sensors have drawn considerable interests due to their super-resolution abilities and high estimation accuracies [1–6]. Over several decades, this topic has been extensively studied and varieties of techniques including multiple signal classification (MUSIC) [7], minimum norm (MN) [8], estimation of the signal parameters via rotational invariance techniques (ESPRIT) [9], subspace fitting (SF) [10], and maximum likelihood (ML) [11] have been developed. In subspace methods, the data matrix or a statistics matrix of the data is normally decomposed into the signal and noise subspaces. Traditionally, this decomposition is usually carried out using an eigen-decomposition (EVD) or a singular value decomposition (SVD) computation, which is computationally intensive and time-consuming. In applications like high-resolution passive sonar systems where the number of sensors is much larger than that of signals [12], the use of such methods is unattractive owing to their intensive computational implementation, and therefore, there is a need for techniques demanding less computations.

Several techniques that seek to determine signal subspace-based estimates without EVD/SVD have been developed. These include the propagator method (PM) [13], the bearing estimation without eigen-decomposition (BEWE) [14], subspace methods without eigen-decomposition (SWEDE) [15], and subspace-based method without eigen-decomposition (SUMWE) [16], in which the signal or the noise subspace is extracted by a linear operator from the array output data and DOA is finally estimated by a cost function similar to MUSIC [17] or root-MUSIC [18]. These methods are found to have significant computational saving over those that explicitly compute EVD or SVD. However, the root mean square error (RMSE) and resolution performances of linear operation-based techniques degrade severely when the signal-to-noise ratio (SNR) is low [19].

Another cross-correlation-based 2D DOA estimation (CODE) algorithm was recently proposed [20] based on an L-shaped array composed of two uniform linear arrays (ULAs). In the CODE method, the noise space is obtained through a linear operation of matrices formed from the forward cross-correlation matrix. It has been shown that CODE can provide good performance with low SNRs and small numbers of snapshots [16]. However, the accuracy of CODE is still inferior to potentially more computationally methods such as MUSIC and root-MUSIC.

The purpose of this paper is to propose an alternate low-complexity subspace-based algorithm without EVD/SVD computation to reduce the computational burden while providing acceptable accuracy for DOA estimation. We present a novel two-step root-MUSIC (TS-root-MUSIC) algorithm to achieve performances closer to the standard root-MUSIC while keeping computational complexity comparable with CODE and being lower than root-MUSIC. In particular, we consider a single ULA which can be possibly regarded as part of a more complex array such as an L-shaped one. We propose to use both the forward and backward cross-correlation matrices [21, 22] and performance enhancement by a further second-step process to improve the estimation accuracy. Numerical simulations are finally provided to illustrate the degree of performance enhancement.

This paper is organized as follows. Section 2 describes the data model of DOA estimation problem using a ULA as well as the background of a standard root-MUSIC algorithm. The proposed TS-root-MUSIC is presented in Section 3, in which the computation of forward and backward cross-correlation matrices, subspace extraction without EVD/SVD, and performance enhancement using the two-step process are discussed in detail. Section 4 summarizes the implementation of the new method and analyzes its computation complexity with comparisons to other methods including root-MUSIC and CODE. Finally, Section 5 conducts numerical simulation results to show that our algorithm costs a less computation burden than root-MUSIC while providing a comparable performance which is much better than CODE.

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

Assume that there are narrowband signals with unknown DOAs simultaneously impinging from far-field on a ULA composed of sensors. Without loss of generality, assume that is an even number, that is, ; the array output vector at time can be written as where is the source waveform vector, is the additive white Gaussian noise (AWGN) vector, and is the array steering matrix. Define , where , is the center wavelength and is the array interval; the steering vector can be written as

The array covariance matrix can be written as where is the source covariance matrix, is the power of AWGN, and is the identity matrix. For full-rank AWGN, the EVD/SVD of matrix can be written in a standard way as where and are the two diagonal matrices of dimensions and , respectively, and and are the so-called signal and noise matrices, which are of dimensions and and contain the eigenvectors relating to the significant and smallest eigenvalues of , respectively. In real-world applications, the theoretical is unavailable, and it is usually estimated by snapshots of array observed data as

Therefore, the EVD of the array covariance matrix is in fact given by

Based on the special geometry of a ULA as well as the orthogonality between span () and span (), the standard root-MUSIC algorithm transforms the tremendous spectral search step involved in MUSIC into a simplified polynomial rooting as [18] where is the projection matrix of the noise subspace. For (7), there are generally pairs of roots that symmetrically distribute around the unit circle. By choosing the roots which are situated the most closely to the unit circle, one can estimate signal DOAs by

Note that the computational complexity of this polynomial rooting step is substantially lower than that of the spectral search [18]. However, as EVD computation requires about flops [20, 23], the complexity of root-MUSIC is still high, especially when large numbers of sensors are used [18].

#### 3. The Proposed Algorithm

##### 3.1. Noise Subspace Extraction without EVD/SVD Computation

As shown in Figure 1, we divide the ULA into two subarrays along its center position without overlapping such that there are elements in each subarray (Note that for the case , the center sensor is shared by the two subarrays and there will be sensors in each subarray.) Thus, both the output vectors of the two subarrays are of dimensions , which can be written as where are the steering matrices of the two subarrays, respectively. Due to the Vandermonde structure, and satisfy where is a exchange matrix with ones on its antidiagonal and zeros elsewhere and is a diagonal matrix, given by