Mathematical Problems in Engineering

Volume 2015, Article ID 235173, 12 pages

http://dx.doi.org/10.1155/2015/235173

## DOA and Noncircular Phase Estimation of Noncircular Signal via an Improved Noncircular Rotational Invariance Propagator Method

^{1}Key Laboratory of Radar Imaging and Microwave Photonics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China^{2}College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 17 January 2015; Accepted 15 April 2015

Academic Editor: Anders Eriksson

Copyright © 2015 Xueqiang Chen 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

We consider the computationally efficient direction-of-arrival (DOA) and noncircular (NC) phase estimation problem of noncircular signal for uniform linear array. The key idea is to apply the noncircular propagator method (NC-PM) which does not require eigenvalue decomposition (EVD) of the covariance matrix or singular value decomposition (SVD) of the received data. Noncircular rotational invariance propagator method (NC-RI-PM) avoids spectral peak searching in PM and can obtain the closed-form solution of DOA, so it has lower computational complexity. An *improved NC-RI-PM* algorithm of noncircular signal for uniform linear array is proposed to estimate the elevation angles and noncircular phases with automatic pairing. We reconstruct the extended array output by combining the array output and its conjugated counterpart. Our algorithm fully uses the extended array elements in the improved propagator matrix to estimate the elevation angles and noncircular phases by utilizing the rotational invariance property between subarrays. Compared with NC-RI-PM, the proposed algorithm has better angle estimation performance and much lower computational load. The computational complexity of the proposed algorithm is analyzed. We also derive the variance of estimation error and Cramer-Rao bound (CRB) of noncircular signal for uniform linear array. Finally, simulation results are presented to demonstrate the effectiveness of our algorithm.

#### 1. Introduction

Over the last several decades, the problem of estimating the direction-of-arrival (DOA) of multiple sources in the field of array signal processing has received considerable attention [1–3]. A variety of DOA estimation algorithms have been developed and applied in many fields, such as mobile communication system, radio astronomy, sonar, and radar [4–13]. Although the maximum likelihood estimator provides the optimum parameter estimation performance, its computational complexity is extremely demanding [8–10]. Simpler but suboptimal solutions can be achieved by the subspace-based approaches, which rely on the decomposition of observation space into signal subspace and noise subspace. For example, multiple signals classification (MUSIC) method [11] and estimation of signal parameters via rotational invariance technique (ESPRIT) algorithm [12, 13] are subspace-based DOA estimation algorithms both well known for their good angle estimation performance. However, conventional subspace techniques necessitate eigenvalue decomposition (EVD) of covariance matrix or singular value decomposition (SVD) of data matrix to estimate the signal and noise subspaces; thus, huge computation complexity will be involved, particularly when the number of sensors is large, such as the large towed arrays in sonar [14, 15]. It is well known that propagator method (PM) does not require EVD of covariance matrix or SVD of received data; thus, the computational load of PM algorithm can be significantly smaller [16]. But the spectral peak searching process is used in conventional PM algorithms [17–20]; in order to save the complexity, rotational invariance PM (RI-PM) algorithms are proposed [21–23], which avoid the spectral peak searching process and can obtain the closed-form solution of DOA.

The binary phase shift keying (BPSK), amplitude modulation (AM), and unbalanced quadrature phase shift keying (UQPSK) modulated signals frequently used in communication systems are noncircular (NC) signals [24]. The statistical parameters of noncircular signal, such as first and second moments, are rotational variant. The noncircularity of signal is investigated to enhance the performance of angle estimation algorithm [25]. Many DOA estimation methods of noncircular signals have been reported, which contain NC-MUSIC algorithms [26–28], NC-ESPRIT algorithms [29–31], and noncircular parallel factor (NC-PARAFAC) algorithm [32]. These algorithms we mention above can estimate more sources and have better angle estimation performance by introducing the noncircularity of signal into the conventional DOA estimation algorithms.

Many DOA estimation algorithms of noncircular signal based on PM have also been studied [33–38]. These noncircular propagator method (NC-PM) algorithms can be divided into two kinds, that is, the noncircular spectral peak searching PM algorithms and noncircular rotational invariance PM (NC-RI-PM) algorithms. In fact, Liu et al. have proposed NC-PM algorithm [34] and real-valued NC-PM algorithm [35]; unfortunately, the spectral peak searching process is involved, which generally requires highly computational complexity. Subsequently, Liu et al. proposed NC-root-PM algorithm [36, 37] by exploiting the uniform distributed characteristic of the array elements, but its complexity is still high. The NC-RI-PM algorithm has been proposed by Sun and Zhou in [38], which has much lower computational complexity than Liu’s algorithm and can obtain the closed-form solution of DOA. However, we find that the reconstructed array output can be optimized, and some array elements in the extended propagator matrix have not been used; thus, the angle estimation performance of NC-RI-PM algorithm can be enhanced further. It is worth noting that the rotational invariance PM (RI-PM) algorithms have also been studied in [39–42], but the noncircularity of signal has not been considered.

The aim of this paper is to develop a computationally efficient parameter estimation algorithm of noncircular signal. We propose an improved noncircular rotational invariance PM (improved NC-RI-PM) algorithm of noncircular signal for uniform linear array, which combine the noncircularity of signal and rotational invariance property between subarrays. Compared with NC-RI-PM algorithm, the proposed algorithm has the following advantages: (1) It has better angle estimation performance than that of NC-RI-PM algorithm. (2) The proposed algorithm has much lower computational complexity. (3) It can estimate elevation angles and noncircular phases with automatic pairing.

The reminder of this paper is organized as follows. In Section 2, we describe the data model for uniform linear array. The improved NC-RI-PM algorithm is presented in Section 3, as well as the computational complexity analysis and comparison. Section 4 derives the variance of estimation error and Cramer-Rao bound (CRB) of the proposed algorithm. Simulation results are presented in Section 5, while the conclusions are shown in Section 6.

*Notations*. , and denote inverse, conjugate, transpose, conjugate-transpose, and pseudoinverse operations, respectively. stands for a diagonal matrix, whose diagonal element is the vector . is identity matrix. is the expectation operator. means to get the phase. denotes the Hadamard product. means the estimated value of . denotes the Frobenius norm.

#### 2. Data Model

We assume that there are uncorrelated narrowband signals impinging on a uniform linear array equipped with antennas as shown in Figure 1, and . We also assume that sources are far away from the array; thus, the incoming waves over the antennas are essentially planes. The noise is additive independent identically distributed Gaussian with zero mean and variance , independent of signals. We denote the elevation angles of sources as , where is the elevation angle of source, .