A Source Localization Based on Signal Attenuation and Time Delay Estimation in Sensor Networks

Transcription

1 A Source Localization Based on Signal Attenuation and ie Delay Estiation in Sensor etwors. Aghasi, M. ashei, and B. ossein Khalaj Abstract he proble o localization in sensor networs, based on both the tie delays in signal reception and signal attenuation is considered in this paper. his approach would enhance the localization perorance copared to when only phase shits resulted by tie delays or only attenuation inoration are used. Considering an attenuation odel or the signal and taing into account the signal reception delays, a cost unction iniization proble is ored. Due to the convergence speed, stability and the advantage o using trust region ethods, a Levenberg-Marquardt algorith is used to ind the location o signal sources. Ipleentation details o the algorith such as closed or equations to calculate the Jacobian at every iteration are also provided in this paper. At the end, soe siulation results would deonstrate the validity o our proposed ethod Index ers Levenberg arquardt algorith, localization, wireless sensors I. IRODUCIO he vast developent in integrated circuits technology oers the possibility o using low cost sall sensors in large sensor networs. Sensors that are capable o counicating aong theselves, either processing the data individually or through a central processing unit. he possibility o having sensors in dierent places, while still counicating as a networ aes the a proising tool or challengingprobles such as localization/tracing o signal sources and particularly acoustic targets. In the past years, a lot o research is devoted to present ethods o estiating the location o signal sources. Usually this localization is based on DOA estiation by processing the phase dierence aong receiving sensors [], [2], easuring the tie delay in receiving a signal at dierent sensors (particularly useul or wideband signals) [3], [4] and easuring the received energy o the signals at dierent locations [5]. For every category o ethods, dierent processing algoriths are proposed. For DOA estiation o narrowband signals authors have proposed using ultiple signal classiication (MUSIC) [6] and axiu lielihood ethod (ML) [7]. For wideband signals two step algoriths o irst estiating the tie delays through ethods such as cross correlation (CC) [8] or generalized cross correlation (GCC) [9] and then least square (LS) localization [0], [4] are proposed. For the sae class o Manuscript received April 23, 202; revised May 3, Aghasi was with Shari University o echnology, ehran, Iran. e is now with ECE departent o Cornell University, Ithaca, Y, USA. (e-ail:ha275@cornell.edu) M. ashei., was with Shari University o echnology, ehran, Iran. e is now with ECE departent o Boston University, Boston, MA 0225USA (e-ail: hashei@bu.edu). B. ossein Khalaj is with the Electrical Engineering Departent, Shari University o echnology, ehran, Iran (e-ail: halaj@shari.edu). signals, Chen et al. in [] proposed using an approxiate axiu lielihood (AML) ethod capable o handling ultiple targets when rather long saples o the signal are available. Localization based on the energy received at dierent sensors is also considered using ethods such as axiu lielihood capable o handling ultiple sources and projection onto convex sets [5], [2]. In the energy based ethods usually the tie delay in signal reception is ignored and based on a propagation odel a priori nown, localization is perored. In this paper, we tacle the proble o localization o ultiple targets by using both the tie delays in reception o the signals and the signal attenuation behavior. Our ethod generalizes the AML approach to coherently beneit the signal attenuation and provide a ore robust algorith, in which the targets should siultaneously satisy the correct tie delays aong the sensors and provide sensible level o attenuation at each sensor. We use a basic odel or the signal attenuation and based on that and considering the tie delays, a axiu lielihood estiation proble is proposed. Copared to the traditional AML ethod, our proposed ethod is ore robust and requires less nuber o saples due to the use o attenuation inoration. he paper is organized as ollows. In Section II, we propose a general or or the received signal at every sensor and later discuss the signal attenuation odel. In Section III, a axiu lielihood estiation o the source location is proposed. Due to stability, convergence speed and beneiting trust region ethods, the Levenberg-Marquardt algorith is considered or solving the resulting least squares iniization proble. Beside ethods o ipleenting the algorith, closed or equations or calculating the Jacobian are provided. In Section IV, we exaine the eiciency o proposed ethod through soe exaples and inally there are soe concluding rears in Section V. II. PROBLEM MODELIG A. Modeling Received Signal Consider acoustic targets o unnown location r K. At a tie rae t, each source is oni-directionally eitting a signal s () t,?? =,. Also S acoustic sensors are placed in nown positions rs, =,..., S, in the sae environent. For every source in the environent, the unction describing the signal attenuation at a point is α( ρ ) ρ the distance o that point to the source itsel is. In general, the signal attenuation ay be a unction o any other paraeters, such as the signal requency, ediu inhoogeneity, etc. For sae o siplicity, in this paper we 423

2 consider it to be an identical or or all sources and solely a unction o the distance to the source. With v being the propagation speed and considering S (? t v) to be the signal o th source, easured away ro that source, the total received signal ro the acoustic sources at every sensor can be odeled as (, ) ( τ, ) x () t = α ρ s t + w t () = ort = nt = s ρ = r r is the distance ro th, S th sensor and τ, ρ, / v source to = is the corresponding tie delay in signal reception. he received signal in () is noralized to each sensor gain, in order to reduce the nuber o notations. Moreover, the ter w () t represents the bacground noise which is considered to be a zero ean white Gaussian with variance σ 2 or the purpose o this paper. Beside the positions r, which are the ain unnowns o the localization proble, the waveor s () t are also unnown. he appearance o τ, (which is related to the unnown quantitie r )as the arguent o an unnown waveor s () t ay be probleatic or any optiization schee perored to solve the localization proble. owever this proble ay be overcoe through applying the discrete Fourier transor to () to extract the tie delays τ, and or an equivalent equation in which the unnown paraeters are separated in individual ters, i.e., j X ( ) = α( ρ, ) exp τ, S( ) + ξ ( = n or = n = s (2) X, S ( ) and ξ ( ) 牋 ere, ( ) are the data, signal and noise spectrus respectively. As stated in [], we would lie to highlight the act that, or (2) to be a valid equivalent or o (), we need n t to be large enough to avoid edge eects and n > n t. As we entioned earlier, the attenuation inoration will help the algorith overcoe the drawbacs with the tie-delay only consideration, however the signal length should still be long enough that the distortion caused by the edge eect does not overcoe the copensation that odeling the attenuation causes. B. Signal Attenuation Model As discussed earlier, our assuption about the attenuation odel in this paper is an identical odel or all sources, which only depends on how ar a point is located ro the acoustic source. Based on (), any signal attenuation odel solely a unction o ρ ay be considered. owever, or the purpose o this paper, we ollow soe recent energy based localization ethods (e.g., see [5], [2]), in which the signal attenuation is considered to be inversely proportional to ρ or basically ( )=. his will be the attenuation odel αρ ρ used to derive later equations in the paper, although with soe slight odiications any other α( ρ ) ay be considered. III. A MAXIMUM LIKELIOOD ESIMAIO OF E UKOWS A. Derivation Based on the attenuation odel proposed, atching o the data spectru with the odel can be written as j X exp S = n ( ) = ρ, τ, ( ) + ξ ( ) Based on the central liit theore ξ ( ) 牋 (3), which is a transored zero ean Gaussian rando variable to the requency doain, will be a coplex zero ean Gaussian itsel. For every requency bin having and a atrix or as X, ( ) = X [ ( ), X S, S ( ) = S [ ( ), S, ξ ( ) = ξ [ ( ), ξ, S, (3) can be written in ( ) = Φ( ) ( ) + ( ) X S ξ (4) j ρ, j ρ, nv nv ρ,e ρ, e Φ ( ) = j ρs, j ρs, nv nv ρs, e ρs, e he negative log-lielihood unction to estiate the unnown paraeters θ including the source positions, and source signal spectrus, is rewritable as (5) θ = argin P P (6) P(0) P() P = (7) Pn ( ) and P( ) = X ( Φ ) S. Siilar to [], based on the signal being real valued, we can only consider n?2 424

3 requency bins and or ˆP with blocs o P() or = n. Moreover,(6) is equivalent to iniizing P ( ) P( )) or every and considering the unnown signal spectru S() the inia should satisy ( P ( ) P( ))/ S ( )= which leads to S( ) = Φ X( ) with Φ ( ) representing the pseudo-inverse o Φ ( ). Replacing the obtained S( )in P ( ), leads to a iniization, only in ters o the source locations, i.e., r = ˆ arginp P ˆ (8) * Jacobian atrix at every iteration and the coluns o the Jacobian atrix are obtained by having P ˆ / and P ˆ / y x, closed ors o which are derived in the Appendix. In the next section, we will bring soe exaples to veriy the validity o the ethod. ( ) = X ( ) Φ ( ) ( ) X ( ) P ˆ? Φ (9) = n In the next section, we provide a ethod o tacling the nonlinear least squares iniization proble in (8). ) A Levenberg-Marquardt Algorith or the Miniization For the purpose o iniizing the cost unction in (8), we suggest using the Levenberg-Marquardt algorith (LMA). Our attention towards this algorith is based on two basic eatures. he irst eature is its convergence properties. LMA is basically considered as a quasi-ewton ethod and provides a rather quadratic convergence, while being stabile [3]. he other eature o this ethod is beneiting ro trust region ethods [3]. In act the cost unction in (8) is not in general convex and ay possess several local inia points. Using a trust region approach would be helpul in sipping soe o the localities and with reasonable initial guesses; there would be a high chance o inding the global iniizer. For a 2D localization proble, as the scenarios in the exaple section, the vector o unnowns would be θ = [ x,..., x, y,..., y ], x and y are the x and y coponents o the position vector r. Clearly, the approach is not liited to 2D Cartesian systes and 3D Scenarios and other coordinate systes ay be considered. For the LMA, which is an iterative algorith we start with a θ (0) as the starting point, which usually contains the best guesses about the unnown values. At every iteration, having θ ( n) already in hand θ ( n + ) can be obtained by solving Fig.. he cost unction corresponding to the traditional AML ethod (top) and our ethod (botto). All nubers are in eters. Fig. 2. Localization or a single target ( n) ( n+ ) ( n) ( λ ) ( ) J J I θ θ J P (0) + = ˆ θ θ θ ˆP is the vertical vector o length n S?2 explained in the previous section and obtained or values ( n) θ at that iteration. Jθ is the Jacobian atrix o size ( n) n S?2 2. he paraeter λ is the daping actor, obtained at every iteration based on the trust region approach [4], [3]. In order to run the algorith we need to now the Fig. 3. Localization or two targets siultaneously 425

4 Fig. 4. he error caused by reducing the signal available saples or both AML and proposed ethod IV. SIMULAIO RESULS In this section we provide soe siulation results to show the perorance o proposed ethod. As an exaple, we consider ive sensors placed in the x y plane as shown in Fig.. A wideband source with center requency 500 z, and 200 z bandwidth located at point (7,6) is considered. he n = saples sapling requency used is 4 Kz and 000 t are taen ro the signal at every sensor. For the algorith used n is taen to be 050 and the SR at every sensor is 5 db. he propagation speed is the speed o sound as v = 345 /s o provide a better understanding o the proble, we have shown the behavior o the cost unction in the sensor-target area in Fig.. he top igure shows the cost unction corresponding to the AML ethod in [], only tie delays are considered and a noralized version o the signal is used. he botto igure shows the cost unction corresponding to our ethod. One ay observe that iposing the additional constraint o attenuation odel has ade the cost unction soother and less luctuations and variations are observable. his is due to the act that ore paraeters (tie delay and aplitude vs tie delay only in the AML ethod) are now involved in the cost unction odeling and it is harder or two neighboring points to dier a lot with respect to both tie delay and attenuation constraints and hence a soother cost unction is resulted. Also, since ore inoration about the signal is used, the localization is iproved as in this setting the AML cost unction taes its inia at ( ) while our cost unction taes its inia at ( ) which is a better estiate o the target. For the purpose o peroring the iniization using the LMA, we need to provide an initial target estiate. Finding good initial estiates o the target can be perored by using a course grid on the searching region. For the tracing probles, however, the localization results in previous tie raes can be used as the initial estiates or the next raes. Despite this, we have initialized the proble ro the point (3, 4) and as shown in Fig. 2, the algorith was still able to ove towards the global iniizer and reach the point (6.99, 5.97). he interediate estiates o the target through the iterations are also shown in that igure. As a ultiple target scenario, we consider the sae proble setting as beore with two targets located at (7, 6) and (8, 3). he initial estiates or the target positions were points (9, 9) and (7, 4). he resulting estiates o the target are shown in Fig. 3, or which the inal results are (7.0, 5.98) and (7.89, 3.00), which are in good atch with the true target locations. We also provide an experient to exaine the perorance o proposed ethod. For this sae we use the single target exaple and start reducing the available saples at the sensors. We always use n = nt + 50 requency bins, while reducing the collected signal saples ro 2000 to 00 and observe the error occurred in target estiations. Fig. 4 shows the relationship between sapling points and the error in source estiates or both the AML and proposed ethod. he igure can well highlight the act that or a certain nuber o signal saples available, using both tie delays and attenuation inoration in the signal enhances the perorance o the localization copared to the case that only tie delays are used. V. COCLUSIO In this paper we proposed a ethod or localization o ultiple acoustic wideband sources based on both signal attenuation and tie delays in sensor networs. he ethod presented taes into account the signal attenuation behavior in the environent and as shown through siulations, provides a ore appropriate cost unction than that o the AML ethod. With the help o the attenuation odel, less nuber o signal saples can be used to peror the localization copared to the case that only signal delays inoration is used. he iniization schee used to solve the resulting nonlinear least square proble is the Levenberg-Marquardt, which uses trust region ethods and less sensitive to local inia points. APPEDIX: JACOBIA CALCULAIO As entioned earlier, in order to ind coluns o the Jacobian, we are required to ind P ˆ?, θ is one o the unnown paraeters x or y. Since P is a vector containing sub-vectors ˆP () or = n, we will only ind P ˆ? ) / and clearly oring Pˆ / would be aligning the corresponding sub-vectors. Fro (9) we have ˆ P ( )/ θ = Φ Φ X θ ). () For sae o siplicity in notations and avoiding the appearance o the requency bin in all the derivations we denote Ψ =? Φ ) and so expanding () yields P ˆ ( ) = ( Ψ Ψ + Ψ Ψ X (2) Ψ We irst consider inding Reerring to Section III, Ψ is siply calculated through Ψ = ( Ψ Ψ? Ψ or in other words 426

5 ( Ψ Ψ Ψ = Ψ (3) aing a derivative ro both sides o (3) results or ( Ψ + ( Ψ Ψ = ( which result in Ψ + ( Ψ Ψ = ΨΨ + Ψ Ψ + Ψ Ψ = (4) Using (4) in (2) and recalling would result Ψ? = Ψ Ψ Ψ?, ˆ P( ) Ψ Ψ Ψ = Ψ + Ψ ( I ΨΨ ) Ψ Ψ X ( (5) which can be written as ˆ ( ) ( ) P Ψ Ψ = I Ψ Ψ Ψ + Ψ ( I ΨΨ ) X ( ) = { Γ+ Γ } X( ) Ψ Γ = ( Ψ Ψ I ) Ψ ow that we have a closed or or calculation o the Jacobian coluns, we can speciically discuss inding Ψ Because o siilar ors, here we only discuss Ψ.Siply x taing a derivative with respect to x in (5) shows that the (, ) eleent o Ψ is related to the (, ) eleent o x Ψ through Ψ xs x 2 j π = δ (, ) + [ Ψ ], x ρ, ρ, nv, δ (, ) ; = = 0; REFERECES [] L. a, arget localization ro bearings-only observations, Aerospace and Electronic Systes, IEEE ransactions on, vol. 33, no., pp. 2 0, [2] G. Carter, Coherence and tie delay estiation, Proceedings o the IEEE, vol. 75, no. 2, pp , [3] M. Brandstein, J. Adcoc, and. Silveran, A closed-or location estiator or use with roo environent icrophone arrays, Speech and Audio Processing, IEEE ransactions on, vol. 5, no., pp , [4] K. Yao, R. udson, C. Reed, D. Chen, and F. Lorenzelli, Blind beaoring on a randoly distributed sensor array syste, Selected Areas in Counications, IEEE Journal on, vol. 6, no. 8, pp , [5] X. Sheng and Y. u, Maxiu lielihood ultiple-source localization using acoustic energy easureents with wireless sensor networs, Signal Processing, IEEE ransactions on, vol. 53, no., pp , [6] R. Schidt, Multiple eitter location and signal paraeter estiation, IEEE transactions on antennas and propagation, vol. 34, no. 3, pp , 986. [7] I. Zisind and M. Wax, Maxiu lielihood localization o ultiple sources by alternating projection, Acoustics, Speech and Signal Processing, IEEE ransactions on, vol. 36, no. 0, pp , [8] G. Carter, Coherence and tie delay estiation: an applied tutorial or research, developent, test, and evaluation engineers, 993. [9] C. Knapp and G. Carter, he generalized correlation ethod or estiation o tie delay, IEEE ransactions on Acoustics, Speech and Signal Processing, vol. 24, no. 4, pp , 976. [0] J. Chen, K. Yao, R. udson,. ung, C. Reed, and D. Chen, Source localization o a wideband source using a randoly distributed beaoring sensor array, pp. 8, 200. [] J. Chen, R. udson, and K. Yao, Maxiu-lielihood source localization and unnown sensor location estiation or wideband signals in the near-ield, Signal Processing, IEEE ransactions on, vol. 50, no. 8, pp , [2] D. Blatt and A. ero, Energy-based sensor networ source localization via projection onto convex sets, Signal Processing, IEEE ransactions on, vol. 54, no. 9, pp , [3] J. Dennis and R. Schnabel, uerical ethods or unconstrained optiization and nonlinear equations. Society or Industrial Matheatics, 996. [4] K. Madsen,. Bruun, and O. I, Methods or non-linear least squares probles, aidreza Aghasi was born in 989 in Isahan, Iran. e received his BSc. in electrical engineering (Counication systes) ro Shari University o echnology, ehran, Iran in 20 and is currently woring toward his PhD degree in electrical and coputer engineering at Cornell University,Ithaca, Y. is current research interest is RF circuit design with a ocus on low phase noise VCO s. e has also conducted research on signal processing and wireless sensor networs. Mr. Aghasi was a candidate to receive the best Bsc. thesis award ro departent o EE at Shari University o echnology in 20. e was naed a Jacobs scholar at Cornell University in 202. Morteza ashei received the B.S. degree in electrical engineering ro Shari University o echnology, ehran, Iran in 20 he conducted research in Advanced Counications Research Institute (ACRI). e is currently a PhD student at Boston University, MA, USA. is research interests include networs perorance evaluation, traic engineering, and wireless networing. e has also conducted research on wireless sensor networs and wireless counications. Mr. ashei was a candidate to receive the best BSc. thesis at EE departent o Shari University o echnology in 20. In all 20 he was awarded the Boston University scholarship. Baba osein Khalaj received his B.Sc. degree in electrical engineering ro Shari University o echnology, ehran, Iran, in 989. and the M.Sc. and Ph.D. degrees in electrical engineering ro Stanord University, Stanord, CA, in 992 and 995 In 995, he joined KLA-encor as a Senior Algorith Designer, woring on advanced processing techniques or signal estiation. Fro 996 to 999, he was with Advanced Fiber Counications and Ianos Counications. Fro 998 to 999, he was the Coeditor o the Special Copatibility Standard Drat or the ASI-E Group. Fro 2006 to 2007, he was a Visiting Proessor with the Centro de Estudios e Investigaciones écnicas de Gipuzoa (CEI), San Sebastian, Spain. e is the author o any papers in signal processing and digital counications. e is the holder o two U.S. patents and was the recipient o the Alexander von uboldt Fellowship during