Wireless Communications and Mobile Computing

Volume 2017, Article ID 1049141, 11 pages

https://doi.org/10.1155/2017/1049141

## Distributed 3D Source Localization from 2D DOA Measurements Using Multiple Linear Arrays

Dipartimento di Elettronica Informazione e Bioingegneria, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy

Correspondence should be addressed to Antonio Canclini; ti.imilop@inilcnac.oinotna

Received 17 May 2017; Accepted 11 September 2017; Published 25 October 2017

Academic Editor: Paolo Barsocchi

Copyright © 2017 Antonio Canclini 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

This manuscript addresses the problem of 3D source localization from direction of arrivals (DOAs) in wireless acoustic sensor networks. In this context, multiple sensors measure the DOA of the source, and a central node combines the measurements to yield the source location estimate. Traditional approaches require 3D DOA measurements; that is, each sensor estimates the azimuth and elevation of the source by means of a microphone array, typically in a planar or spherical configuration. The proposed methodology aims at reducing the hardware and computational costs by combining measurements related to 2D DOAs estimated from linear arrays arbitrarily displaced in the 3D space. Each sensor measures the DOA in the plane containing the array and the source. Measurements are then translated into an equivalent planar geometry, in which a set of coplanar equivalent arrays observe the source preserving the original DOAs. This formulation is exploited to define a cost function, whose minimization leads to the source location estimation. An extensive simulation campaign validates the proposed approach and compares its accuracy with state-of-the-art methodologies.

#### 1. Introduction

The problem of acoustic source localization with microphone arrays has been an active topic of research in the last thirty years [1]. Several are the applications that benefit from this technology, among which video-conferencing [2, 3], audio-surveillance [4], and so forth are worth mentioning.

In the literature, it has been proposed to use either compact or distributed arrays. Compact arrays are advantageous since their deployment is easy. As an example, in a video-conferencing system, a compact array can be located in the proximity of the camera to steer it towards the detected speaker. Unfortunately, compact arrays pose some limits in terms of localization accuracy. Indeed, for far sources, only the direction of arrival (DOA) can be accurately estimated, while the range estimate is generally unreliable [1]. On the other hand, distributed arrays envision the presence of multiple sensors, each equipped with one or more microphones, located around the volume of interest [5–10]. With this configuration, the source is observed from different angles, with benefits in terms of localization accuracy. Until recently, the cost of the deployment of the sensors and the cabling made this kind of solutions cumbersome and possible only in specific ad hoc applications. In the last few years, however, the advent of wireless sensor networks made this kind of solutions attractive for a wider range of applications.

When each sensor is equipped with multiple microphones, the direction of arrival of the source can be measured internally to each sensor. The source is then localized by a central node, which combines measurements coming from the individual sensors. The advantage of DOAs with respect to other measurements (e.g., time differences of arrival or times of arrival) lies in the fact that each sensor transmits a single measurement to the central node, independently from the number of microphones.

Source localization from multiple DOAs can be brought to the problem of triangulating the estimated DOA lines [11, 12]. Following this approach, 3D localization is typically accomplished by combining multiple 3D DOAs measured at sensors with noncollinear microphone arrangements. For instance, multiple spherical arrays are used in [13]. A 3D DOA is represented by a pair of angles, which denote the azimuth and the elevation of the source with respect to the sensor position. Therefore, measuring a 3D DOA involves a two-dimensional search space. As robust DOA estimation algorithms are generally based on grid-search solutions (e.g., beamforming [14], steered-response-power [15]), the inherent power requirement may be an issue for their implementation at local sensors. For this reason, many works limit source localization to 2D (planar) geometries, assuming that all the sensors and the source lie on the same plane. This is typically accomplished through the triangulation of multiple 2D DOAs, as done, for instance, in [8, 12, 16]. From a geometrical standpoint, a 2D DOA is measured in the plane containing the array and the source and consequently consists of a single angle. The search space is therefore one-dimensional, and simple array geometries can be adopted for this task. For instance, using a linear array, the DOA can be estimated by evaluating an objective function in the range , suitably sampled according to the desired angular resolution. Moreover, in order to achieve a prescribed resolution, a smaller number of microphones are required compared to that needed for 3D DOA measurements. For these reasons, 2D DOA measurements are more affordable for low-power sensing devices, as typically required in designing wireless sensor networks.

In this manuscript, we propose a technique that localizes a source in 3D from the combination of 2D DOAs, measured using multiple linear microphone arrays. Differently from state-of-the-art methods exploiting 2D DOAs, we remove the requirement of working in a planar geometry. Specifically, we do not pose any constraint about the positions of the source and of the arrays in the 3D space. The proposed technique proceeds in two steps. First, each array measures the 2D DOA of the source referred to a local plane, determined by the lying line of the array and the location of the source. Using the concept of* equivalent arrays*, the array positions are roto-translated into equivalent ones all lying in the same plane containing also the source. Each equivalent array position is found so that the original DOA is preserved, as well as its distance from the source. Once this transformation has been accomplished, in the second step the source can be localized in a 2D geometry. In order to do this, we take advantage of the Ray Space [17], in which DOAs are interpreted as acoustic rays originating from the source and impinging on the acoustic centers of the sensors. Acoustic rays are parameterized by the parameters of the line on which they lie. A cost function is defined, whose minimization gives us the source location.

The proposed technique is advantageous with respect to state-of-the-art methods for 3D source localization using DOAs, which, to the best of our knowledge, all rely on 3D measurements. Being based on simpler 2D DOA measurements, indeed, the proposed method requires a reduced number of microphones for each sensor of the network, with obvious advantages in terms of hardware cost (microphones and analog-to-digital converters). For the same reasons, power consumption at each sensor turns out to be significantly reduced, as the beamforming operation leading to the 2D DOA estimate is accomplished on a one-dimensional search space. Finally, it is worth noticing that, exploiting a simple geometry such as that of a linear array, the manufacturing costs are much smaller than those required to realize more complex structures (e.g., spherical and cylindrical arrays).

The remainder of the manuscript is structured as follows. Section 2 gives some background information concerning the Ray Space representation and the localization of acoustic sources in 2D. Section 3 introduces the theoretical framework for the proposed approach, with particular reference to the equivalent arrays. Section 4 addresses the problem of source localization using the equivalent arrays. Section 5 is devoted to the validation of the proposed technique. Finally, Section 6 draws some conclusions.

#### 2. Background

##### 2.1. Ray Space Representation

Consider an acoustic source in a 2D scenario. Its Cartesian coordinates are . In geometrical acoustics, it can be described as the set of all acoustic rays that originate from it. Each ray can be parameterized with the line on which it lies. The homogeneous representation of all the lines passing through is This is equivalent to the vector form The parameter vector is homogeneous; that is, any scaling , , represents the same line. Consequently, the homogeneous coordinates form a class of equivalence in a two-dimensional projective space, defined in [18] as the* projective Ray Space*.

##### 2.2. DOAs in the Ray Space

Let us now assume that the acoustic source is located in the far field with respect to the center of a microphone array ; that is, the length of the array is much smaller than the curvature of the wavefront. In this context, the spatial information of the source can be described in terms of its direction of arrival (DOA), that is, the angle of propagation of the wavefront. Source and array location are related to the DOA by The DOA defines the acoustic ray that joins the source position and the observation point, which turns to be oriented as . In the Ray Space, the DOA is thus represented as the acoustic ray passing through and directed as , whose parameters are

##### 2.3. 2D Source Localization in the Ray Space

We now consider the problem of localizing the acoustic source, by combining a set of DOAs measured by a distributed network of linear microphone arrays. With reference to Figure 1, the acoustic center of the th array is at , , and the orientation of the line on which it lies is given by the unit vector . The vector forms the angle with the -axis. The DOA of the source, as measured with respect to the array line, is represented by the angle . The DOA is more conveniently represented in the global reference frame as .