A Graphical Model Approach to Source Localization in Wireless Sensor Networks

Transcription

1 A Graphical Model Approach to Source Localization in Wireless Sensor Networks Manish Kushwaha, Xenoon Koutsoukos Institute or Sotware Integrated Systes (ISIS) Departent o Electrical Engineering and Coputer Science Vanderbilt University, Nashville, TN 37235, USA Eail: anish.kushwaha@vanderbilt.edu Abstract Collaborative localization and discriination o ultiple acoustic sources is an iportant proble in Wireless Sensor Networks (WSNs). Localization approaches can be categorized as signal-based and eature-based ethods. The signal-based ethods are not suitable or collaborative localization in WSNs because they require transission o raw acoustic data. In eature-based ethods, signal eatures are extracted at each sensor and the localization is done by ultisensor usion o the extracted eatures. Such ethods are suitable or WSNs due to their lower bandwidth requireents. In this paper, we present a eature-based localization and discriination approach or ultiple haronic acoustic sources in WSNs. The approach uses acoustic beaor and Power Spectral Density (PSD) data ro each sensor as the eatures or ultisensor usion, localization, and discriination. We use a graphical odel to orulate the proble, and eploy axiu likelihood and Bayesian estiation or estiating the position o the sources as well as their undaental and doinant haronic requencies. We present siulation and experiental results or source localization and discriination, to deonstrate our approach. In our siulations, we also relax the source assuptions, speciically the haronic and onidirectional source assuptions, and evaluate the eect on localization accuracy. The experiental results are obtained using otes equipped with icrophone arrays and an onboard FPGA or coputing the beaor and the PSD. Keywords: Acoustic source localization, Wireless sensor networks, Feature-level usion, Probabilistic graphical odel, Bayesian estiation, Maxiu likelihood estiation. 1 Introduction Acoustic source localization is an iportant proble in any diverse applications such as ilitary surveillance and reconnaissance, underwater acoustics, seisic reote sensing, and environental onitoring [1, 2, 3]. Recently, innovative applications such as sart video-conerencing [4], audiovideo sensor usion and target tracking [5, 6] have also been proposed to utilize source localization. Traditional acoustic source localization ethods were developed or wired sensor networks [7]. In Wireless Sensor Networks (WSNs), collaborative source localization is used to estiate the positions o ultiple sources by usion o observations ro ultiple sensors. There are two broad classes o ethods or collaborative source localization. The irst class o approaches, where the estiation is done by usion o the raw sapled signals, is called signal-based or signal-level usion. The 1

2 second class o approaches, where signal eatures are extracted ro raw data at each sensor and the estiation is done by usion o the extracted eatures, is called eature-based or eature-level usion. The signal-level usion ethods are not suited or WSNs because they require transission o the raw signal, which is costly due to liited bandwidth and power. On the other hand, the eature-level usion ethods are appropriate or WSNs due to their lower bandwidth and power requireents. In this paper, we present a eature-level usion approach to collaborative localization and discriination or ultiple haronic sources in WSNs. Source discriination involves spatial discriination where the goal is to separate the sources in space, and requency discriination where the goal is to separate the sources in requency. We use beaors and Power Spectral Densities (PSDs) as the signal eatures. The advantage o using the beaor over signal energy is that the beaor captures the angular variation o signal energy, which results in better localization resolution, and hence better spatial source discriination. Assuing haronic sources, the use o PSD as another signal eature allows requency discriination. Advances in sensor network hardware and FPGA integration has allowed us to ipleent real-tie algoriths or coputing such eatures. Furtherore, the counication bandwidth available in WSNs is suicient to support wireless transissions o such eatures [6]. We use a graphical odel to orulate the proble, and eploy Maxiu Likelihood (ML) and Bayesian estiation or estiating the position o the sources as well as their undaental and doinant haronic requencies. Directed graphical odels, which are generalization o Bayesian networks, are directed graphs in which nodes represent rando variables, the directed edges represent causality between rando variables, and the (lack o) edges capture conditional independence o rando variables [8]. We represent the unknown source locations and requencies as hidden state variables, and the acoustic eatures as observable variables (or observed data). In directed graphical odels, the edges are directed ro the hidden state variables to the observed data. Directed graphical odels require generative odels that describe the observed data in ters o the process that generated the, and the hidden state variables. We present generative odels or the beaor and PSD data that describe the in ters o a generative process and the unknown source locations and requencies. In our approach, the solution to the collaborative localization and discriination proble is divided into two steps as source separation and source localization. The idea is to separate the sources in the requency doain using the PSD data ro the sensors, and then use the separated sources or localization and discriination. We analytically show that source separation is independent o source localization, as long as the sources and the sensors are stationary. We develop a ML estiation ethod or source separation and use Bayesian estiation or localization. We use Markov Chain Monte Carlo (MCMC) ethods, speciically Gibbs sapling and slice sapling [9] or ipleenting both ML and Bayesian estiation. The advantages o the two step approach instead o joint estiation are twoold. First, estiation in two steps has lower coputational coplexity than joint estiation. Second, the variances o the likelihood unctions or source separation and source localization are signiicantly dierent. In context o Monte Carlo ethods, joint estiation ay cause slower convergence, and require a large nuber o saples. We present siulation results or ultiple source localization in a grid sensor network. We study three siulation scenarios where (1) we increase the nuber o sources, (2) we increase the average source SNR o two sources present in the sensing region, and (3) we increase the separation between the two sources. Our results show that as the separation between sources increases, the algorith is able to achieve higher localization accuracy, coparable to single source localization. We present evaluation o the localization accuracy when the assuptions or the acoustic sources are relaxed; speciically the haronic and onidirectional source assuptions. The localization accuracy degrades graceully when source haronicity is decreased. We also present evaluation 2

3 o the localization accuracy with PSD data copression. As expected, the localization accuracy iproves as ore PSD data is available. Finally, we ipleent the eature extraction algoriths on an FPGA chip onboard icaz sensor nodes, and conduct outdoor experients with real acoustic sources. Outdoor experiental results reinorce the siulation results. For saller source separations the average localization error reains low but the algorith is not able to disabiguate the two sources. For larger separations, the localization error is decreased. The rest o the paper is organized as ollows. Related work is present in Section 2. In Section 3, we present the acoustic source odel, signal propagation odel and eature extraction algoriths. Section 4 describes the graphical odel. Sections 5 and 6 describe the source separation and source localization algoriths, respectively. In Section 7, we describe Gibbs sapling and slice sapling. We present results or various siulation scenarios in Section 8, and the outdoor experient setup and results in Section 9. We conclude in Section Related Work Signal-based ethods or acoustic source localization typically ake use o Tie Delay o Arrival (TDOA) and Direction o Arrival (DOA). An overview o theoretical aspects o TDOA-based acoustic source localization and beaoring is presented in [10], along with a localization algorith based on ML estiation. For ultiple acoustic sources, an Approxiate Maxiu Likelihood (AML) algorith based on alternative projection ethod is also presented. An epirical study o collaborative acoustic source localization based on an ipleentation o AML is shown in [3]. Aong eature-based ethods, Energy-Based Localization (EBL) ethods utilize signal energy as the eatures. Least-squares orulations or EBL have been presented in [11, 12]. A ML orulation with capability or ultiple source localization is presented in [13]. They use a ultiresolution search algorith and an expectation-axiization (EM) like iterative algorith or estiation. We use the beaor instead o the signal energy as the eature. The advantage is that the beaor captures the angular variation o signal energy, which results in better localization resolution, and hence better source discriination. The classical approaches to ultiple target tracking include data association-based approaches such as Multiple Hypothesis Tracking (MHT) [14] and data association ilters [15, 16]. These approaches use a set o exclusive and exhaustive hypotheses either associating easureents with the targets and clutter, called target-oriented ethods, or associating targets with easureents, called easureent-oriented ethods. Probabilities are coputed or each hypothesis and the ost probable hypotheses are used to copute target estiates. The nuber o hypotheses is cobinatorial in the nuber o targets and easureents, as well as in tie. In data association-based approaches, the easureents are noise-corrupted sensor readings related to the state o a target, such as range and/or aziuth ro a sensor, etc. The easureents are usually not raw data points, but rather the outputs o signal processing and detection subsystes [16]. The sensor odel assues that each easureent (or detection) corresponds to a single target (i.e. each easureent originated either due to a single target or due to noise). Also, each target generates a single easureent and the easureent due to one target is not aected by the presence o other targets. In other words, a easureent or a target would be sae regardless the presence o other targets in the scene. Due to these assuptions, the data-association based approaches are not able to odel target interaction and ixed easureents, which results in unresolved targets [17]. The proble o ixed easureents and unresolved targets is even ore signiicant in acoustic sensors because the acoustic source signals ro ultiple targets are additive. 3

4 Due to the act that easureents are not raw data points, but the outputs o signal processing and detection subsystes, the data association-based tracking can be categorized as high-level (or decision-level) usion. In high-level usion, discriinating inoration is lost, especially or sensing odels where raw data is a ixture o the signals originating ro ultiple targets. Alternate approaches can utilize low-level (or signal-level) usion, or ediu-level (or eature-level) usion. Signal-level usion ethods are not suitable or WSNs because they require transission o the raw signal, which is costly due to liited bandwidth and power. Feature-level usion ethods include signal processing and eature extraction algoriths that extract inorative eatures ro the raw data, which are counicated to the sensor usion node. These ethods require explicit target interaction odels and observation odels that describe the generation o eatures. Feature-level usion ethods are appropriate or WSNs due to their lower bandwidth and power requireents, at the sae tie, they aintain suicient discriinating inoration. More recent approaches to ultitarget tracking include joint tracking using Bayesian inerence where the quantity o interest is the joint ultitarget state, which is the concatenation o individual target states [18]. Joint Bayesian inerence has the advantages o providing a recursive solution with arbitrary target dynaic odels and observation odels. Several approaches based on Bayesian estiation [19, 20] and graphical odels [5, 21] have been also proposed. Sequential Monte Carlo (SMC) ipleentations or Bayesian estiation and ultitarget tracking are presented in [22, 23]. A Bayesian approach or tracking the DOA o ultiple targets using a passive sensor array is presented in [19]. A Bayesian approach or ultiple target detection and tracking, and particle ilter-based algoriths are proposed in [20]. A graphical odel based approach or audiovisual object tracking that uses audio and video data ro a icrophone pair and a caera is presented in [5]. A graphical odel orulation or sel-localization o sensor networks using a technique called nonparaetric belie propagation is presented in [21]. The eature-based localization approach in this paper also uses a graphical odel that odels ultiple target interaction and ixed easureents through generative odels or the acoustic eatures. We use MCMC ethods or source separation and localization, and we analytically show that source separation is independent o source localization. An alternative to Bayesian statistics or ultiple target tracking is Finite Set Statistics (FISST). The Bayesian raework treats the states and observations as realizations o rando variables. In FISST raework the ultitarget state and ultiple observations are treated as inite sets. The systeatic treatent o ultisensor ultitarget tracking using rando set theory and FISST is presented in [24]. The FISST Bayes ultitarget recursion is generally intractable. An approach to approxiate the ultitarget Bayes recursion by propagating the Probability Hypothesis Density (PHD) o the posterior ultitarget state is proposed in [25]. This strategy is siilar to the constant gain Kalan ilter that propagates the ean o the posterior single-object state. The PHD recursion still involves ultiple integrals with no closed ors in general. Several SMC ipleentations o the PHD ilter are proposed in [26, 27]. Like hypothesis-based tracking approaches, the FISST based tracking also assues one-to-one association between targets and easureents, which results in siilar liitation as or the hypothesis-based tracking approaches. 3 Acoustic Source Localization & Discriination We consider a WSN o K acoustic sensors in planar ield and M ar-ield stationary acoustic sources coplanar with the sensor network. The objective is to estiate the 2D position o all the sources and discriinate the, both in requency and space, using the received acoustic signals at the sensors. To localize the acoustic sources, we assue each sensor node is equipped with an array o N ic icrophones. The acoustic wave ront incident on the icrophone array is assued to be planar 4

5 or ar-ield sources. Each sensor receives an acoustic signal that is a cobination o source signals. The sensors run signal processing algoriths to copute the beaor and PSD eatures. In the rest o the section, we describe the source assuptions, source odel, signal propagation odel and signal processing algoriths or eature extraction. 3.1 Acoustic Source Model The ain assuptions ade or the acoustic sources are: (1) onidirectional and stationary point sources, (2) eitting stationary signals, (3) the source signals are haronic, and (4) the crosscorrelation between two source signals is negligible copared to the signal autocorrelations. Haronic signals have a undaental requency, also called the irst haronic, and other higher-order haronic requencies that are ultiples o the undaental requency. The energy o the signal is contained in these haronic requencies only. The haronic source assuption is satisied by a wide variety o acoustic sources [28]. In general, any acoustic signal originating due to the vibrations ro rotating achinery will have a haronic structure. The state or the th acoustic source is given by: (1) the position x () = [ x (),y ()] T (), (2) the undaental requency ω, and (3) the energies [ ] T in the haronic requencies ψ () = ψ () 1,ψ () 2,,ψ () H where H is the nuber o haronic requencies. In practice, soe o the assuptions ay not be always true. For exaple, the engine sound o a vehicle ay not be onidirectional and will be biased toward the side closer to the engine. The physical size o the acoustic source ay be too large to be adequately odeled as a point source or sensors very close to the source. In an outdoor environent, strong background noise, including wind gusts, ay be encountered during operation. Perhaps the ost restrictive assuption is that the source signals are haronic. In addition to the haronic coponents, the engine sound signal ay contain other requency coponents, which when not accounted or, ay cause localization to deteriorate. In Section 8, we present epirical evaluation o localization accuracy when the source assuptions are relaxed, speciically the haronic and onidirectional sources assuptions, and it is shown that the localization accuracy degrades graceully. 3.2 Signal Propagation Model The intensity o an acoustic signal eitted oni-directionally ro a point sound source attenuates at a rate that is inversely proportional to the distance ro the source [13]. The discrete signal received at the p th icrophone on a particular icrophone array is given by r p [n] = M =1 d 0 x p x () s() [n τ () p ]+w p [n] (1) or saples n =1,,L,whereL is the length o the acoustic signal, M is the nuber o sources, w p [n] is white Gaussian easureent noise such that w p [n] N(0,σw), 2 s () [n] is the intensity o the th source easured at a reerence distance d 0 ro that source, τ p () is the propagation delay o the acoustic signal ro the th source to the p th icrophone, and x p denote the icrophone position. We deine the ultiplicative ter in Equation (1) as the attenuation actor, λ () p,given by λ () d 0 p = x p x (). 5

6 3.3 Acoustic Features The two acoustic eatures used or eature-level usion are beaor and PSD. The details o the eature extraction algoriths are given below. Details related to an FPGA ipleentation are given in Section Beaor Beaoring is a signal processing algorith or Direction-o-Arrival (DOA) estiation o a signal source using an array o icrophone [10]. In a typical delay-and-su single source beaorer, the 2D sensing region is discretized into directions, or beas as α = i 2π Q,wherei =0,,Q 1andQ is the nuber o beas. The beaorer coputes the energy o the reconstructed signal at each bea direction. This is achieved by delaying and suing the individual icrophone signals. The delayed and sued signal is given by r[n] = N ic p=1 r p [n + t pq (α)] (2) where α is the bea angle, r p [ ] is the received signal at the p th icrophone, q is the index o a reerence icrophone, N ic is the nuber o icrophones, and t pq (α) is the relative tie delay or the p th icrophone with respect to the reerence icrophone q, givenby t pq (α) =d pq cos(α β pq ) s /C where d pq and β pq are the distance and angle between the p th and q th icrophones, and s and C are signal sapling rate and speed o sound, respectively. The bea energy is given by B(α) = L r[n] 2 = n=1 L n=1 [ Nic p=1 r p [n + t pq (α)] Bea energies are coputed or each o the beas, and are collectively called the beaor. The bea with axiu energy indicates the DOA o the acoustic source. In case o ultiple sources, there ight be ultiple peaks where the axiu peak would indicate the DOA o the highest energy source. Figure 1(a) shows a beaor or two acoustic sources Acoustic PSD PSD estiation is a signal processing algorith or estiating the spectru o the received acoustic signal, which describes how the power o the signal is distributed with requency [29]. We copute the PSD as the agnitude o the Discrete Fourier Transor (DFT) o the signal P (ω) =Y (ω) Y (ω) where Y (ω) =FFT(r, N FFT ) is the DFT o the signal r[n], N FFT is the length o the transor, and Y (ω) is the coplex conjugate o the transor. For real-valued signals, the PSD is real and syetric; hence we need to store only hal o the spectral density. In our ipleentation, we copress the PSD data or wireless transission by storing and sending the requency-power pairs, (ω j,ψ j ), or N PSD requencies with the highest power values. Figure 1(b) shows an acoustic PSD estiate or a received signal when two haronic sources are present. ] 2 6

7 source 1 source 60 2 both sources angle (degree) (a) PSD (db) requency (Hz) (b) Figure 1: (a) Acoustic Beaors; The beaors or single sources clearly show peaks at the source location but the beaor when both sources are present does not show two peaks. (b) Power spectral density (PSD); the highest PSD values are shown as epty circles. The PSD is copactly represented as pairs o the highest PSD values and corresponding requencies. 4 Graphical Model Overview Probabilistic graphical odels provide a systeatic ethodology to handle uncertainty and coplexity in real systes. They are playing an increasingly iportant role in the design and analysis o achine learning, bio-inoratics, audio processing and iage processing algoriths [8, 30]. In a probabilistic graphical odel, each node represents a rando variable (or a group o rando variables), and the edges express probabilistic relationships between these variables. The lack o an edge between nodes represents conditional independence. The graph structure captures the way in which the joint distribution over all the rando variables can be actorized into a product o local unction deined on nodes and their iediate neighbors (a subset o rando variables). Hence, probles involving coputation o quantities related to the joint probability odel can be proitably cast within a graph theoretic raework. In particular, the underlying structure o the graph is closely related to the underlying coputational coplexity o an inerence proble [8]. The graphical odels can be categorized as undirected and directed graphical odels. Undirected 7

8 graphical odels capture correlation between rando variables, while directed graphical odels capture causality between variables. In directed graphical odels, also called Bayesian networks, the edges are directed ro hidden variables to observed variables. The directed graphical odels require generative odels that describe the observed data in ters o the process that generated the, and the hidden variables. The odel shown in Figure 2 is an exaple o a directed graphical odel. We use this graphical odel to orulate the source separation and localization o ultiple sources proble. We use plate notation to represent the repetition o the rando variables [31]. The plate index, M on the upper plate represents repetition o the hidden variables or M sources, while the index, K on the lower place represents repetition o the observed variables or K sensors. In the igure, the nodes with clear x ω ψ M K θ k λ k K B k P k Figure 2: Graphical odel. background denote hidden state variables; x (),ω (),ψ () denote source position, undaental requency and haronic energies or the th source, respectively. The nodes with shaded backgrounds denote observed variables; B k and P k denote the beaor and the PSD received at k th sensor, respectively. Finally, the nodes with dotted outlines denote unctions o rando variables, or auxiliary rando variables that capture the unctional dependence o the observed variables on the hidden variables. The two auxiliary variables shown in the graphical odel are the angle θ () k and the attenuation actor λ () k. These variables will be utilized in the generative odels or the observed variables. We peror ultiple source localization and discriination in two steps. First, we use the PSD data only to separate the sources, which in our proble, reers to separating the PSDs o the sources. For haronic sources, estiation o undaental requencies is suicient or source separation, because all the doinant requencies in the signal are ultiples o the undaental requency. An ML estiation ethod is proposed in the next section or undaental requency estiation. It is shown that the ML estiate is independent o the source location, which is intuitive because the doinant requencies in the source signal are independent o the source location, as long as the source and the sensor are stationary. In the second step, we use the beaor data and the separated source PSDs to localize all the sources. We chose to peror ultiple source localization in two steps instead o joint estiation because o the ollowing two reasons. First, estiation in two steps has lower coputational coplexity than joint estiation. In the context o Monte Carlo ethods, let the nuber o saples needed be exponential in state diension, N D α,whered 1isthe state diension and α 1isthe exponential actor. The joint estiation o two state variables with diensionality D 1 and D 2 would 8

9 require N joint (D 1 + D 2 ) α saples, while separate estiation would require N separate D α 1 + Dα 2 saples. For D 1,D 2,α 1, one can show that N joint N separate. Second, the variances o the likelihood unctions or source separation and localization are signiicantly dierent. In Monte Carlo context, joint estiation ay cause slower convergence, and require a large nuber o saples. Moreover, as entioned earlier, the ML estiate or source undaental requencies is independent o the source locations, urther supporting the two step process o source separation and source localization. 5 Source Separation In this section, we present the irst step o our approach, which is source separation and requency discriination. We use the PSD data only to separate the sources.we propose an ML estiation ethod or source separation. In ML estiation, we need the data likelihood unction or the PSD data, which requires a generative odel. We begin by presenting the generative odel and the likelihood unction or the PSD data. We also present a result showing that the likelihood unction at the ML estiate o haronic energies is independent o the source positions. Finally, we present the ML estiate or source undaental requency. 5.1 Generative Model or PSD Data For haronic sources, the PSD can be given by P () s (ω) = H h=1 where =1,,M are source indices, ω is the requency, ω () ψ () h δ(ω hω () ) (3) is the undaental requency, ψ () h is the energy in the h th haronic, H is the nuber o haronics, and δ( ) is the Dirac delta unction. Using Equation (3), we derive a generative odel or the PSD data received at a sensor node. The ollowing proposition states the generative odel or the PSD data. 1 Proposition 1 For an arbitrary nuber o acoustic source signals, the power spectral density o the signal received at a sensor is given by P(ω) = M M =1 n=1 λ () λ (n) ( P () s cos(φ () (ω) Φ (n) (ω)) (ω)p (n) (ω) s ) 1 2 (4) where M is the nuber o sources, λ () is the attenuation actor, and Φ () (ω) is the phase spectral density, which is given by Φ () (ω) =φ () x () x s ω/c where φ () is the phase o the source signal, x () and x s are the positions o the source and the sensor, respectively. 1 The proo o this and all other propositions are given in the appendix. 9

10 The expression in Equation (4) can be approxiated using the ollowing observation. Since we do not aintain the phase o the signal in the source odel (see Section 3), we assue all the phases to be norally distributed with equal ean. The expected value o the cosine o the dierence o two norally distributed angles is one, i.e. E[cos(Φ i Φ j )] = 1. Thereore, Equation (4) can be approxiated as [ M ( ] 2 1/2 P(ω) λ () P (ω)) () (5) =1 5.2 Data Likelihood & ML Estiate Using Equation (5), the negative log-likelihood or PSD data at the kth sensor is deined as l k (Ω, Ψ, X) = 1 σp 2 P k (ω) P k (ω) 2 dω (6) ω where P k (ω) istheobservedpsdatthekth sensor, and [ ] T Ω = ω (1),,ω(M) Ψ= [ψ (1),, ψ (M)] T [ ] T ψ () = ψ () 1,,ψ () H X = [x (1),, x (M)] T Assuing a discrete requency variable, Equation (6) can be rewritten as l k (Ω, Ψ, X) = 1 σ 2 P ω j P k (ω j ) P k (ω j ) 2 The likelihood unction or the PSD data at the ML estiate o haronic energies is independent o the source positions, as stated in the ollowing proposition. Proposition 2 The likelihood or the PSD data at the ML estiate o haronic energies is the likelihood that is given by l k (Ω, Ψ ML, X) =l k(ω, X) = (P k (ω j )) 2 = l k(ω ) (7) ω j H(Ω ) where H is the set o all haronic requencies or all sources H(Ω )= [ ω (), 2ω (), and, the likelihood is a unction o the source undaental requencies only, and is independent o the source positions. ] T 10

11 Hence, according to Proposition 2, source separation can be perored independent o source localization. The ull negative log-likelihood or all sensors, l (Ω ) is deined as l (Ω )= 1 K K l k(ω ) Thus, the ML estiation o the undaental requencies can be obtained by iniizing l (Ω ) ˆΩ ML k=1 =arginl 1 (Ω )=argin Ω Ω K K k=1 ω j H(Ω ) (P k (ω j )) 2 (8) Note that Equation (8) is not an explicit expression or Ω since the set H is a unction o Ω on the righthand side o this equation. This otivates the use o an iterative ethod or ML estiation. WeuseaMonteCarloethoddescribedinSection7. 6 Source Localization Source localization is perored by Bayesian estiation in the graphical odel shown in Figure 2, and taking the axiu a-posteriori (MAP) estiate o the source positions. The posterior, p(x B) o the source positions at the ML estiates or source undaental requencies and haronic energies given the beaor data ˆΩ ML p(x B) K p(b k X, k=1 ML ˆΩ, ˆΨ ML )p(x) where p(b k X,, ˆΨ ML ) is the likelihood unction or beaor data, X represent joint state or all sources, and B represent the beaors or all sensors. The likelihood unction requires a generative odel or the beaor data. In this section, we present the generative odel and three interediate results pertaining to the odel. Finally, we present the likelihood and MAP estiation. 6.1 Generative Model or Beaor We start by developing a generative odel or a beaor or a two-icrophone array, singlesource case (Proposition 3). We will show that the beaor or an arbitrary icrophone array (Proposition 4) and an arbitrary nuber o sources (Proposition 5) can be coposed ro the siple two-icrophone array, single-source case. Proposition 3 Consider a icrophone pair separated by distance d and the angle between the x- axis and the line joining the icrophones is β. For an acoustic source at angle θ and range r with power spectral density P (ω), the beaor B at the icrophone pair is given by B(α) =2λ 2 (R ss (0) + R ss (κ α )) + 2R η (0) (9) where R ss (τ) =FFT 1 (P (ω)) or τ [, + ] is the autocorrelation o the source signal, R ss (0) is the signal energy, R η (0) is the noise energy, λ is the attenuation actor, and κ α = d(cos(α β) cos(θ β)) s /C, whereα [0, 2π] is the bea angle, s and C are sapling requency and speed o sound, respectively. 11

12 For an arbitrary icrophone-array, the generative odel can be extended as ollows. Proposition 4 For an arbitrary icrophone-array o N ic icrophones, the beaor is expressed in ters o pairwise beaors as B(α) = B i,j (α) N ic (N ic 2)(R η (0) + λ 2 R ss (0)) (10) (i,j) pa where pa is the set o all icrophone pairs, R ss (0) is the signal energy, R η (0) is the noise energy, λ is the attenuation actor, and B i,j is the beaor or the icrophone pair (i, j) (Equation (9)). For an arbitrary nuber o acoustic sources, the generative odel is given by the ollowing proposition. Proposition 5 For an arbitrary nuber o uncorrelated acoustic sources M, the beaor is expressed in ters o single source beaors as B(α) = M B (α) N ic (M 1)R η (0) (11) =1 where R η (0) is the noise energy and B is the beaor or th acoustic source (Equation (10)). Finally, the generative odel or arbitrary icrophone array and arbitrary nuber o sources can be obtained by substituting Equations (9) and (10) into Equation (11), which gives B(α) =2 M =1 λ ()2 (i,j) pa R () ss (κ α)+n ic 6.2 Data Likelihood & MAP Estiate M =1 Using Equation (12), the negative log-likelihood or beaor data is given as ln p(b k X) =l k (X) = 1 σ 2 B The MAP estiate o the source positions is given by λ ()2 R () ss (0) + N icr η (0) (12) B k (α) B k (α) 2 ˆX MAP =argaxp(x B) (13) X Since the generative odel or beaor in non-linear, an exact ethod or state estiation in Equation (13) is not possible and we use Monte Carlo ethod described in the next section or state estiation. 7 Monte Carlo Estiation Markov Chain Monte Carlo ethods are a class o Monte Carlo ethods or sapling coplex probability distributions based on constructing a Markov chain that has the desired distribution as its equilibriu distribution. MCMC approaches are so-naed because they use the previous saple values to randoly generate the next saple value, generating a Markov chain (as the transition α 12

13 probabilities between saple values are only a unction o the ost recent saple value). The state o the chain ater a large nuber o steps is then used as a saple ro the desired distribution. The quality o the saple iproves as a unction o the nuber o steps. The MCMC ethods are ore eicient, especially or probles with high-diensional state-space, than sequential Monte Carlo (SMC) ethods, also called particle ilters [23]. This is due to the act that the saples in SMC ethods are drawn independently, while saples in MCMC are drawn ro a Markov chain. I the desired distribution is highly localized in a high-diensional state space, ost o the independent saples drawn by SMC ethods would have low probability. On the other hand, ater a suicient nuber o steps, saples drawn by MCMC ethods would, in act, be ro the desired distribution. The Metropolis-Hastings (MH) algorith is the earliest o the MCMC ethod [32]. The MH algorith generates a Markov chain using a proposal density which depends on the current saple. The proposed saples are accepted as part o Markov chain according to a acceptance-rejection rule. Another popular MCMC ethod called the Gibbs sapler is very widely applicable to a broad class o Bayesian probles [33]. The Gibbs sapler is a special case o Metropolis-Hastings sapling wherein the proposed saple is always accepted. This results in lesser rejected saples, hence better eiciency. Gibbs sapling, however, requires that all the conditional distributions o the target distribution can be sapled. In this paper, we use Gibbs sapling or estiation. Gibbs sapling algorith works on the idea that while the joint probability distribution is too coplex to draw saples ro directly, the univariate conditional distributions the distribution when all but one o the rando variables are assigned ixed values are easier to saple. We denote the state vector as X = [ x (1),x (2),,x (D)] t, where D is the nuber o state variables. The joint density p(x t X t 1,Y t ) is sapled using Gibbs sapler by sequentially sapling univariate conditional densities given by x (k,j) t p(x (j) X (k, j) t,x t 1,Y t ) (14) where k is the index [ o the saple, j =1,,nis the index o ] state variable currently being sapled, t and X (k, j) t = x (k,1) t,,x (k,j 1) t,x (k 1,j+1) t,,x (k 1,D) t is the set o all state variables except x (j). In any cases, the univariate conditional distribution can be arbitrary and the choice o onediensional sapling algorith to saple ro the univariate distribution deterines the speed and convergence o the Gibbs sapler. We select slice sapling or its robustness in paraeters such as step size and applicability toward non-log-concave densities, which is the case in our proble due to ultiodal probability distributions [34]. The pseudo code in Algorith 1 presents the Monte Carlo ethod or source separation and localization using Gibbs sapling and slice sapling. At tie t = 0 (line 1), the Gibbs sapling algorith is initialized with source undaental requencies and source locations. For each tie t>0, we peror ML estiation or source separation (lines 3 10) and Bayesian estiation or source localization (lines 11 19). The Gibbs sapler draws N saples (line 4 & 12) ro the target distribution by sequentially drawing ro univariate conditional distributions (lines 5 7 & 13 16). Note that slice sapling is used or univariate sapling (lines 6 & 14 15). Notation ro Equation (14) is used in lines 6 & The state or k th saple is shown in lines 8 & 17. For ML estiation, the saple with iniu negative-log-likelihood is selected as the estiate (line 10). For Bayesian estiation, the saple with axiu a posteriori is selected as the state estiate (line 19). 8 Siulation Results Typically, localization o an acoustic source in WSNs is perored by the sensors that are close to the source because the signal-to-ratio (SNR) is lower or arther sensors. For this reason, we assue 13

14 Algorith 1 Monte Carlo source separation and source localization algorith 1: At t = 0, initialize Gibbs sapler (Ω,0, X 0 ) 2: or t>0 do 3: %%% Source Separation (MC-ML Estiation) 4: or k =1,,N do 5: or =1,,M do 6: saple ω (k,) 7: end or[ 8: Ω (k),t =,t ω (k,1),t p(ω () ],,ω (k,m),t Ω (k, ),t, Ω,t 1,P t ) 9: end or ML 10: ML estiate, ˆΩ,t =argin (k) Ω l (Ω (k),t ),t 11: %%% Source Localization (MC Bayesian Estiation) 12: or k =1,,N do 13: or =1,,M do 14: saple x (k,) t 15: saple y (k,) 16: end or[ 17: X (k) t = t x (k,1) t 18: end or 19: MAP estiate, 20: end or p(x () X (k, ) t p(y () X (k, ) t,y (k,1) t ˆX MAP t,,x (k,m) t =argax X (k) t, X t 1,B t,, X t 1,B t, ],y (k,m) t ML ˆΩ,t ) ˆΩ ML,t ) p(x (k) t X t 1, B) that even in a large sensor network, a source will be surrounded by a sall nuber o sensors that will participate in the localization o that source. 8.1 Setup and Paraeters In siulations, we consider a sensor network o 4 acoustic sensors arranged in a grid o size 10 5, wherein each sensor can detect all the sources. We siulate the sources according to the acoustic source odel in Section 3, siulate the data according to the eature extraction algoriths in Section 3, and inally copare the output o source localization against the ground truth. The perorance o the approach is easured in ters o the localization error, which is deined as the root ean square (RMS) position error averaged over all the sources E = 1 M M x () x () =1 where M is the nuber o source, and x () and x () are the estiated and the ground truth positions or the th source, respectively. Table 1 shows the paraeters used in the algorith. 8.2 Frequency Discriination Figure 3(a) shows the PSD data likelihood or two sources. The data likelihood is highly ultiodal but localized. The data likelihood near the true undaental requency values is higher than the likelihood or other requencies. Figure 3(b) shows the localized nature o the PSD data likelihood. The data likelihood is centered around the true undaental requency with a standard deviation 14

15 Table 1: Paraeters used in siulations Sapling requency ( s ) 100kHz Speed o sound (C) 350 /sec Audiodatalength(tie) 1sec Maxiu haronic requency (ω ax ) 1000Hz SNR (db) 25 Nuber o beas 36 Size o Fourier transor (N FFT ) 4000 Nuber o Gibbs saples 40 o less than 0.01 Hz. Due to this localized nature o the data likelihood, we can discriinate undaental requencies as close as 0.1 Hz. (a) (b) Figure 3: (a) PSD data likelihood, (b) Frequency discriination. 8.3 Localization and Spatial Discriination We study three siulation scenarios. In the irst scenario, we increase the nuber o sources present in the sensing region gradually to see the eect on localization accuracy. In the second scenario, we increase the average source SNR o two sources present in the sensing region. In the third scenario, we increase the separation between two sources present in the sensing region to evaluate spatial discriination. Figure 4(a) shows the localization error or the irst scenario when the nuber o sources is increased ro 1 to 4. The localization error increases approxiately exponentially with the nuber o sources. Figure 4(b) shows the average localization error or the second scenario when source SNR or the two sources is increased ro 7dB to 52dB. As expected, the localization error decreases with increasing SNR and reains approxiately constant above 25dB. Figures 4(c) and 4(d) show the localization error or the third scenario when the source separation between the two sources is increased ro 0.1 to 8. For sall source separations (0.1 and 0.2), the localization error is 15

16 1.4 Localization Error 0.7 Localization Error average RMS position error () average RMS position error () nuber o target SNR (db) (a) (b) 0.18 Localization Error 140 Localization Error average RMS position error () localization error (%) target separation () (c) target separation () (d) Figure 4: Localization error with (a) Source density, (b) Source SNR, and (c) Source separation. (d) Localization error as a percentage o source separation. o the sae order as the separation. This indicates that the two sources cannot be disabiguated at such separation. For higher source separation (above 0.5), the localization error is a sall raction o the separation distance. This indicates that the two sources are successully localized and discriinated. In act, or larger source separation (above 5), the average localization error or the two sources is the sae as that o the single sources. 8.4 Relaxation o Source Assuptions We evaluate the localization accuracy when the assuptions or the acoustic sources are relaxed; speciically the haronic and onidirectional sources assuptions Haronicity Haronicity o a signal represents the degree o acoustic periodicity in the signal. In this analysis, we deine signal haronicity as the ratio o signal energy in the haronic requencies over the total 16

17 signal energy = ω H P (ω) ω P (ω) (15) Acoustic signals originating ro rotating achinery will have high haronicity close to unity, while signals due wind or a white noise source will have low haronicity. The localization accuracy o our approach degrades graceully when signal haronicity is decreased. Figure 5(a) and 5(b) show the localization error or single source and two sources, respectively, with signal haronicity. The localization error decreases as the signal haronicity is increased. Hence, the signal haronicity coputed using Equation (15) can be also used as an indicator o conidence in the localization result. (a) (b) Figure 5: Localization error with signal haronicity or (a) Single Source, (b) Two Sources Directivity Directivity o an acoustic source is a easure o its directional characteristics. Directivity indicates how uch signal energy is directed toward a speciic area copared to the total signal energy being transitted by the source. In this analysis, we express 2D source directivity in ters o a directionality coeicient that governs the attenuation o the signal energy with the angle. The directional signal attenuation is given by ( ) β 2+cosφ λ φ = 3 where β is the directionality coeicient and φ is the direction. The value o directionality coeicient is zero (β = 0) or onidirectional sources (i.e. the signal attenuates uniorly in all directions). Figure 6(a) shows the localization error or two sources when the directionality o the sources is increased. The localization error or beaor-based localization is not aected or directionality coeicient as high as 1.0, ater which the error increases rapidly. As copared to energy-based localization, beaor-based localization is able to cope with higher directionality. It is expected that i we increase the nuber o sensors in beaor-based localization, the eect o source directionality can be urther reduced. 17

18 (a) (b) Figure 6: Localization error with (a) Source directionality, (b) Aount o PSD data available. 8.5 PSD Data Copression We also present epirical analysis or the eect o PSD data copression to the localization accuracy by increasing the N PSD paraeter presented in Section As expected, the localization accuracy iproves when ore PSD data is available. Figure 6(b) shows the localization error or two sources as a unction o size o PSD data available to the usion algorith. As expected, the localization error decreases as ore PSD data is ade available to the base station. The accuracy is poor and degrades rapidly or saller PSD data size. 9 Outdoor Experients We ipleented the beaoring and PSD estiation algoriths described in Section 3 on a Xilinx XC3S1000 FPGA based icaz sensor otes (see Figure 7(a)). The outline o the overall design is as ollows. First, the FPGA collects saples o the signal received at the icrophones in FIFO buers and perors beaoring and PSD estiation. Then the FPGA stores the results, i.e. beaoring energies and N PSD highest values o the PSD, in registers easily accessible ro the icaz ote. Once these registers are read the ote transits their value to the base station where urther processing and sensor usion takes place. The block diagra o the FPGA design is shown in Figure 7(b), where the upper hal (above the dashed line) represents the beaoring- and the botto hal shows the PSD estiation coponent. Beaoring coponent utilizes 166 sec o audio data each cycle, while the PSD estiation coponent utilizes 1 sec o data with 75% overlap. The angular resolution o beaoring is 10 degrees while requency resolution o PSD estiation is 1 Hz. The PSD estiation coponent returns 30 PSD values. Beaoring Coponent. The entire FPGA application runs at 20 MHz and the Analog to Digital Converters (ADCs) are sapling at 1 MSPS. The beaoring coponent uses all our icrophone channels. The 8-bit ADC values are irst downsapled by a actor o 10 and stored in a FIFO buer. FIFO buers are realized as circular buers using 2-Kbyte ebedded block RAMs that store saple values ro last 20 s. Read access to the FIFO buers and write access to 18

19 ADC1 10 FIFO ADC2 10 FIFO ADC3 10 FIFO abs( )² DMPX Beaoring Energy (BE) Registers ADC4 10 FIFO Beaoring Coponent PSD estiation Coponent 250 FIFO Beaorer Control Logic (BCL) PSD Control Logic (PCL) FFT abs( )² Peak Sort (PS) PSD Index and Magnitude (PIM) Registers Restart Interrupt (a) (b) Figure 7: (a) Acoustic sensor node with 4 icrophone. (b) Block diagra o Beaoring- and PSD estiation coponents realized on FPGA. the Beaoring Energy (BE) registers are controlled by the Beaorer Control Logic (BCL). Eleents in the FIFOs are selected or read by a siple state achine that resides in the BCL. The state achine iterates through 36 states each corresponding to a speciic direction and accesses the FIFO eleents based on a look-up-table that contains the delay inoration or the dierent angles. The accessed values o the our FIFOs are sued, squared and accuulated in the corresponding BE register in one FPGA clock cycle or each direction. Once the energies or all the 36 directions have been calculated, BCL waits or the next saple to arrive and starts this procedure over. This procedure is repeated 2 14 ties which takes approxiately 160 s and ensures that the 32-bit BE registers do not overlow. Ater the last cycle, a trigger signal is generated or the PSD estiation coponent and the beaoring coponent is halted. PSD Estiation Coponent. The PSD estiation coponent uses the saples ro only one channel. The signal is irst re-sapled at 4 khz. For the deciation process, a 600-tap poly-phase ilter with 2 khz cut-o requency is used. The deciated 8-bit saples are then ed into a 4-Kbyte FIFO buer which allows to store up to 1 s saple history. In response to a trigger event received ro the beaoring coponent the FIFO content is loaded into the 4096 point FFT odule. Only the agnitude inoration o the irst 2048 FFT results is orwarded to the Peak Sort (PS) odule. The PS odule then selects and stores 30 eleents with largest agnitudes and their corresponding indices. Ater all the FFT saples are processed an interrupt signal is generated or the icaz ote, which in return reads all the register values through an I2C bus, transits the values through the radio and restarts the beaoring coponent. The PSD estiation tie is negligible copared to the I2C and radio transers, which take approxiately 90 s together. The beaoring process in conjunction with the PSD estiation and the data transers takes roughly 250 s, which results in an approxiately 4 Hz update rate at the base station. 19

20 Resource Utilization. The resources used by the beaoring- and PSD estiation coponents and the overall design (including all driver odules) are shown in Table 2. The utilization rates o the dierent FPGA coponents copared to available resources ound in the Xilinx XC3S1000 FPGA prove the easibility o this application, but also points out its liitations. Since 75% o block RAMs are already used, eory resources are likely to becoe a bottleneck when using larger data sets (FIFO sizes). Beaoring PSD Estiation Overall Design Flip Flops 1,258 (8%) 2,018 (13%) 3,843 (8%) 4 input LUTs 1,398 (9%) 2,578 (16%) 6,860 (44%) Block RAMs 8 (33%) 9 (38%) 18 (75%) Hardware ultipliers 1(4%) 6 (25%) 7 (29%) Table 2: FPGA resource utilization o the Beaoring-, PSD Estiation odules and the overall design in absolute nubers and relative to the available resources ound in the Xilinx XC3S1000 FPGA. Experients using an Outdoot Deployent We deployed a sensor network o 3 icaz-based acoustic sensor nodes in an equilateral triangle o side length (15t). Figure 8(a) shows the experiental setup and the location o the sources. We collected the sensor data and ran the algorith oline. Figure 8(b) shows the localization error with source separation. The results ollow the siilar trend as that in Figure 4(c). For saller source separations, the average error reains low but the algorith is not able to disabiguate the two sources. For larger separations, the localization error decreases. (a) average RMS position error () source separation () (b) Figure 8: (a) Outdoor experiental setup. Source 1 is kept at the sae location while source 2 is placed at dierent locations. (b) Localization error with source separation. 10 Conclusion In this paper, we proposed a eature-based usion ethod or localization and discriination o ultiple acoustic sources in WSNs. Our approach used beaors and PSD data ro each sensor. The approach utilized a graphical odel or estiating the source positions and the undaental 20