International Journal of Antennas and Propagation

Volume 2015 (2015), Article ID 348636, 9 pages

http://dx.doi.org/10.1155/2015/348636

## Reduced-Dimension Noncircular-Capon Algorithm for DOA Estimation of Noncircular Signals

^{1}College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China^{2}Luoyang Optoelectro Technology Development Center, Luoyang 471000, China^{3}National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China

Received 2 September 2015; Revised 18 November 2015; Accepted 22 November 2015

Academic Editor: Wei Liu

Copyright © 2015 Weihua Lv 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

The problem of the direction of arrival (DOA) estimation for the noncircular (NC) signals, which have been widely used in communications, is investigated. A reduced-dimension NC-Capon algorithm is proposed hereby for the DOA estimation of noncircular signals. The proposed algorithm, which only requires one-dimensional search, can avoid the high computational cost within the two-dimensional NC-Capon algorithm. The angle estimation performance of the proposed algorithm is much better than that of the conventional Capon algorithm and very close to that of the two-dimensional NC-Capon algorithm, which has a much higher complexity than the proposed algorithm. Furthermore, the proposed algorithm can be applied to arbitrary arrays and works well without estimating the noncircular phases. The simulation results verify the effectiveness and improvement of the proposed algorithm.

#### 1. Introduction

Direction of arrival (DOA) estimation is a hot topic in the array signal processing field and has been widely used in communication, radar, sonar, and medical image [1–4]. Classical DOA estimation algorithms include multiple signal classification (MUSIC) [5], estimation of signal parameters via rotational invariance technique (ESPRIT) [6–8], propagator method [9], and the Capon [10]. Besides, compressive sensing (CS) [11] and Bayesian compressive sensing (BCS) [12] have recently been used to solve the problem of DOA estimation, and they have an advantage of not requiring knowledge of the number of impinging signals.

To improve the DOA estimation performance, the noncircular property of incoming signals has been considered in [13–22]. In wireless communications, the noncircular signals have been extensively used, for example, the binary phase shift keying, amplitude modulation, and unbalanced quadrature phase shift keying [22]. If , , and , then is a noncircular signal [13–15]. This statistics redundancy can be properly exploited to enhance the DOA estimation performance. In general, we use the array output and its conjugation to extend the data model and array aperture. A noncircular MUSIC (NC-MUSIC) algorithm was proposed in [14] for the DOA estimation of the noncircular signals. In order to avoid the peak search in NC-MUSIC, a polynomial rooting NC-MUSIC (NC-Root-MUSIC) was presented in [15]. NC-ESPRIT algorithms were proposed in [16, 17] for DOA estimation without spectrum search. Real-valued implementation of unitary ESPRIT (NC-Unitary-ESPRIT) for noncircular sources was presented in [18], and it has a low complexity. Besides, a noncircular propagator method (NC-PM) for direction estimation of noncircular signals was proposed in [19], which has better angle estimation performance than PM in [9]. Based on the parallel factor (PARAFAC) technique, a noncircular PARAFAC (NC-PARAFAC) algorithm was proposed in [20] to obtain the two-dimensional (2D) DOA estimation of the noncircular signals for arbitrarily spaced acoustic vector-sensor array. Moreover, a two-dimensional direction-finding for noncircular signals using two parallel linear arrays via the extended rank reduction algorithm was presented in [21].

Many DOA estimation algorithms mentioned above require the prior knowledge of the number of sources or need to estimate the number of sources by information theory algorithm, matrix decomposition algorithm, smoothed rank algorithm, or Gerschgorin disks algorithm [23–26]. Notably, the Capon method can work well without information of the number of sources [10]. CS method and BCS method work well without estimating the number of the sources, but the noncircular property is not considered. By combining the Capon method and the noncircular property, a noncircular Capon (NC-Capon) algorithm was proposed in [27] for the DOA estimation of noncircular signals, but it needs an exhaustive two-dimensional (2D) search over the regions of both DOA and noncircular phase.

In this work, we will propose a reduced-dimension NC-Capon (RD-NC-Capon) algorithm for the DOA estimation of noncircular signals. The proposed algorithm, which only requires one-dimensional search, can avoid the high computational cost within the two-dimensional NC-Capon (2D-NC-Capon) algorithm [27]. The angle estimation performance of the proposed algorithm is much better than that of the conventional Capon algorithm and very close to that of the two-dimensional NC-Capon algorithm. Moreover, the proposed algorithm can be applied to arbitrary arrays and works well without estimating the noncircular phases. Numerical simulations verify the improvement and effectiveness of the proposed algorithm.

The remainder of this paper is structured as follows. Section 2 introduces the data model. Section 3 proposes the RD-NC-Capon algorithm. Section 4 gives the performance analysis. In Section 5, simulation results are provided to show the effectiveness, while the conclusions are drawn in Section 6.

*Notations*. Lowercase (capital) bold symbols denote vector (matrix). , , , , and denote complex conjugation, transpose, conjugate-transpose, inverse, and pseudoinverse operations, respectively. stands for a diagonal matrix whose diagonal is a vector . is to get the real part of the complex. presents the statistical expectation. denotes the element of the matrix . is Hadamard product. stands for an identity matrix and is a zero matrix with .

#### 2. Data Model

As shown in Figure 1, we consider a linear array consisting of omnidirectional sensors and select the first sensor as the referenced one. The vector denotes the displacement between the sensor and the referenced one, and . We assume that there are far-field, narrow-band signals impinging on the linear array from different angles . The received signal of the array can be expressed by vector [28]:where with and being the wavelength. is the narrow-band noncircular signal vector and denotes the additive white Gaussian noise.