International Journal of Antennas and Propagation

Volume 2015 (2015), Article ID 783467, 15 pages

http://dx.doi.org/10.1155/2015/783467

## Structure-Aware Bayesian Compressive Sensing for Near-Field Source Localization Based on Sensor-Angle Distributions

Center for Advanced Communications, Villanova University, Villanova, PA 19085, USA

Received 31 December 2014; Revised 25 March 2015; Accepted 30 March 2015

Academic Editor: Wen-Qin Wang

Copyright © 2015 Si Qin 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 technique for localization of narrowband near-field sources is presented. The technique utilizes the sensor-angle distribution (SAD) that treats the source range and direction-of-arrival (DOA) information as sensor-dependent phase progression. The SAD draws parallel to quadratic time-frequency distributions and, as such, is able to reveal the changes in the spatial frequency over sensor positions. For a moderate source range, the SAD signature is of a polynomial shape, thus simplifying the parameter estimation. Both uniform and sparse linear arrays are considered in this work. To exploit the sparsity and continuity of the SAD signature in the joint space and spatial frequency domain, a modified Bayesian compressive sensing algorithm is exploited to estimate the SAD signature. In this method, a spike-and-slab prior is used to statistically encourage sparsity of the SAD across each segmented SAD region, and a patterned prior is imposed to enforce the continuous structure of the SAD. The results are then mapped back to source range and DOA estimation for source localization. The effectiveness of the proposed technique is verified using simulation results with uniform and sparse linear arrays where the array sensors are located on a grid but with consecutive and missing positions.

#### 1. Introduction

Localization of emitters is an important topic with many applications in radar, sonar, and wireless communications. The majority of research work in array signal processing has been focused on far-field signals; that is, the source signals are assumed to be located far from the receiver. Accordingly, the propagation waves assume a planar wavefront, making source localization solely based on the direction-of-arrival (DOA) of the sources. However, this characterization becomes invalid when sources are located in the near-field region. In this case, each source is characterized by both DOA and range that vary with sensor positions. To solve near-field source localization problems, subspace-based techniques, for example, MUSIC, were extended to a two-dimensional field [1]. However, this entails a high computational load required for both DOA and range searching.

To decouple these two variables of the DOA and range for simplified computations, a more accurate quadratic approximation of the wavefront, in lieu of the linear approximation, can be made with respect to the sensor positions [2]. Consequently, the phase delay is no longer linear with the sensor positions but is decided by the DOA and range of the sources jointly. A number of methods have been proposed to jointly estimate the DOA and range information for near-field source localization. For instance, algorithms based on high-order statistics were proposed in [2–4]. The work in [2] exploits the rotational invariance in certain cumulant-domain signal subspace for range and bearing estimation. The technique in [3] applies a symmetric uniform linear array (ULA) and uses eigenvalues together with the corresponding eigenvectors of two properly designed matrices to jointly estimate signal parameters. As such, it avoids any spectral peak searching. Two orthogonally polarized sensors were utilized in [4] to improve performance. These methods, however, do not meet the desired performance and require certain array properties, for example, a ULA with an interelement spacing less than half a wavelength. In practical applications, such approaches are very costly due to receiver hardware and computational complexity.

Sparse arrays [5–7] use fewer elements to achieve the same array aperture of fully populated arrays. They have similar mainbeam properties and, therefore, provide similar performance in terms of angular accuracy, resolution, and detection of targets close to interference directions, all being achieved with reduced size, weight, power consumption, and cost. An algorithm was proposed in [8] to cope with the near-field source localization problem using sparse arrays. However, it relies on the assumption that the sparse array configuration must be symmetric. In addition, similar to [2–4], the second-order Taylor approximation of the range between the source signals and the sensors is assumed to decouple the two variables. As a result, the performance is compromised when the sources are placed close to the array.

In this paper, we propose an alternative approach using sparse arrays by mapping the source range and DOA information into sensor-dependent phase progression. The sensor-angle distribution (SAD) is used to represent the phase progression in the joint space and spatial frequency domain with a signature that resembles those of instantaneously narrowband frequency-modulated (FM) signals in the joint time-frequency domain [9, 10]. In this respect, the signal processing algorithms introduced for quadratic time-frequency analysis can be readily applied for the processing of the SAD for near-field source localization. It is important to note that, while the SAD was originally developed in the form of spatial Wigner distribution [9, 10], which is effective only for second-order (linear FM) spatial frequency signatures, we consider it in the general form within Cohen’s class with a proper time-frequency kernel for improved sensor-angle relationship [11]. The proposed technique permits the sensor-dependent phase progression as a high-order polynomial phase signal (PPS) to account for close sources, whereas the coefficients are estimated from the SAD signature with simple least squares fitting, which does not require parameter searching.

Another important contribution of this paper is the consideration of the source localization problem in a sparse array configuration where the array sensors are located in a uniform linear grid but with missing positions. As such, conventional time-frequency analysis tools become infeasible due to the undesirable artifacts caused from missing sensor entries [12]. Instead, we use the emerging compressive sensing (CS) [13] techniques to reconstruct the SAD in the joint space and spatial frequency domain. The problem is considered in the Bayesian compressive sensing (BCS) framework that generally achieves desired performance and enables flexible utilization of various signature structures through the application of proper priors [14–17]. In particular, we use the structure-aware BCS, which was developed for effective reconstruction of sparse time-frequency distributions of FM signals, to exploit the sparsity and continuity of the spatial frequency signatures in the SAD domain. In doing so, the isolated or sporadic entries in the reconstructed SAD caused by the presence of missing sensors and measurement noise can be significantly suppressed, leading to more reliable SAD reconstruction and spatial frequency estimation. The obtained results are then mapped back to the estimation of source range and DOA information for effective source localization. Compared to the approach proposed in [8] using a sparse symmetric array, the proposed technique offers higher flexibility and robustness to array design and improves the source localization accuracy when the sources are close to the array and thus the phase exhibits a high-order polynomial phase relationship with the sensor positions.

The rest of the paper is organized as follows. Section 2 introduces the signal model of near-field localization based on a third-order approximation. The SAD and its sparse reconstruction are described in Section 3, and the structure-aware Bayesian compressive sensing technique applied for sparse SAD reconstruction and spatial frequency estimation is described in Section 4. Simulation results are provided in Section 5 to numerically evaluate and compare the performance. Such results reaffirm and demonstrate the effectiveness of the proposed technique. Section 6 concludes this paper.

*Notations*. We use lower-case (upper-case) bold characters to denote vectors (matrices). In particular, denotes the identity matrix. denotes complex conjugate, whereas and , respectively, denote the transpose and conjugate transpose of a matrix or vector. represents a diagonal matrix that uses the elements of as its diagonal elements. implies the Euclidean () norm of a vector. expresses the probability density function (pdf) and denotes the conditional pdf of random variable , given other parameters. Beta() computes the beta function for corresponding elements and . describes that random variable follows a complex Gaussian distribution with mean and variance . Furthermore, is the Dirac delta function of .

#### 2. Signal Model

For the simplicity of presentation, the signal model is presented for the single-source case, but the proposed technique can be straightforwardly extended to the multiple source problems and simulation results are provided for two source scenarios.

Consider a near-field narrowband source impinging on a uniform or sparse linear array with elements, as depicted in Figure 1, where . Note that we assume to be an odd integer without loss of generality, but the problem can be similarly formulated for an even value of . Examples of even values of are provided in Section 5. Let the index of the reference element be 0, and the sensors are located at , where for , and is the unit interelement spacing. To avoid spatial ambiguity, is usually taken as half-wavelength, denoted as . The source is located with a range and azimuth angle with respect to a reference sensor. Note that the array may be asymmetric; that is, it is not necessary to assume , for .