Performance Comparison of Nonlinear Filters for Indoor WLAN Positioning

Transcription

1 Perfrmance Cmparisn f Nnlinear Filters fr Indr WLAN Psitining Hui Wang Andrei Szab Jachim Bamberger Learning Systems Infrmatin and Cmmunicatins Crprate Technlgy Siemens AG Munich Germany {hui.wang.ext.andrei.szab.jachim.bamberger}@siemens.cm Dietrich Brunn Uwe D. Hanebeck Intelligent Sensr-Actuatr-Systems Labratry (ISAS) Institute f Cmputer Science and Engineering Universitat Karlsruhe (TH) Germany brunn@ira.uka.de. uwe.hanebeck@ieee.rg Abstract-Indr WLAN psitining shuld be mdeled as a nnlinear and nn-gaussian dynamic system due t the cmplex indr envirnment radi prpagatin and mtin behaviur. The aim f this paper is t analyze different filtering strategies fr real life indr WLAN psitining systems. The perfrmance criteria fr the cmparisn are the mean f lcalizatin errrs and cmputatinal cmplexity. Three nnlinear filters are analyzed: Furier density apprximatin (FF) particle filter (PF) and gridbased filter (GF) which are representatives fr deterministic and randm density apprximatin appraches. Our experimental results help t chse the apprpriate filtering techniques under different resurce limitatins. Keywrds: Indr Psitining Nnlinear Filtering. 1. INTRODUCTION Indr WLAN psitining system has attracted a lt f research interests frm bth academia and industry in recent years [1] [2] [3]. Mathematically it can be frmulated as a state estimatin prblem. A system mdel describes the state evlutin with time i.e. the mtin f mbile devices. A measurement mdel describes the nisy received signal strength (RSS) bservatins. Due t the cmplexity f indr envirnment indr radi prpagatin is highly nnlinear which results in a nnlinear measurement mdel. In additin the mtin f a mbile device cannt be mdeled linearly in general. Therefre we regard the indr WLAN psitining system as a nnlinear and nn-gaussian system. Bayesian filtering prvides a general recursive framewrk t estimate the psterir prbability density functin f the state given all the available bservatins. But its ptimal analytic slutin with respect t an ptimality criterin is tractable nly in sme special cases. Fr instance if the system has a linear structure and the psterir density is kept as Gaussian the ptimal slutin with respect t minimal psterir mean square errr can be derived using the well-knwn Kalman filter [4]. If the state space is discrete and has a limited size the grid-based methd is ptimal [5]. Fr mst nnlinear and nn-gaussian systems like indr WLAN psitining system it is impssible t get the exact analytic slutin due t the fact that the cmplexity f the psterir prbability increases after each step f the recursive prcess. T slve this prblem varius subptimal nnlinear filters have been prpsed. These filters apprximate either the system and measurement mdel r the prbability density functin. The first family f filters including the extended Kalman filter (EKF) [5] and its variants apprximate the nnlinear mdel by its lcal linearizatin. The ther family cnsists f numerus nnlinear filters that aim t apprximate the density functin by a number f parameters. Fr instance grid-based methds use unifrmly sampled grid pints t apprximate the density functins. And Unscented Kalman filter (UKF) [6] matches the first tw mments fthe psterirdistributin density fthe state vectr cnditined n the bservatins by using sample pints f the Gaussian-Hermite Quadrature apprximatin t the integrals. Besides particle filters [5] randmly generate a cllectin f particles t d the same wrk. Recently researchers als represent the density functin by the linear cmbinatin f kernel functins e.g. Gaussian mixtures [7] Furier densities [8] Dirac mixtures [9] and s n. The perfrmance f varius nnlinear filters are clsely tied t the prperties f specific applicatins. The system and measurement mdel the nise characteristics and the system requirements fr speed r memry cnsumptin are the key factrs t decide which filter is mst suitable. In this paper we cmpare the perfrmance f three nnlinear filters namely the filter based n Furier density apprximatin (FF) particle filter (PF) and grid-based filter (GF) fr ur indr WLAN psitining system. We chse these three filters as candidates because they all can handle nnlinear and nn-gaussian systems with arbitrary cmplexities by apprximating density functins with any required accuracy. The key perfrmance criteria we use are the mean lcalizatin errr and cmputatin time which reflect the accuracy and cmputatinal efficiency respectively. We evaluate the filters with bth simulated and real measurements aiming t find the mst apprpriate ne fr practical indr WLAN psitining systems. Our experiments indicate that particle filters are the mst cmputatinally efficient. Furier-based filters are superir with respect t using the fewest cmpnents t apprximate the density functins accurately. The remainder f this paper is rganized as fllws: In Sectin II three nnlinear filters t be evaluated are briefly described. In Sectin III we intrduce the usage f nnlinear filter in indr WLAN psitining systems. The perfrmance

2 f different filters in ur testbed are given and discussed in Sectin IV. Finally Sectin V cncludes the paper. II. DESCRIPTION OF NONLINEAR FILTERS The discrete-time stchastic dynamic system with additive nises is usually mdeled by Frmula (1) mdels the system prcess which updates the current state vectr x k given an input Uk and system nise Wk. Frmula (2) describes the measurement prcess which relates the state x k+1 t the bservatin Yk+1 crrupted als by nise Vk. The Bayesian apprach prvides a recursive way t estimate the hidden state fdynamic systems with the abve frm. It has als tw steps: predictin step and update step as If+l (Xk+l) = J/f (xk+lixk Uk) fk (Xk) dxk (3) fk+l (xk+d = ft+l (Yk+llxk+d ff+l (Xk+l) (4) Ck where fk+l (Xk+l) is the predicted density at time k. f'{+l (Xk+ll xk Uk) is the transitin density which is given by f'{+l (Xk+ll xk Uk) = Ii: (Xk+l - ak (Xk' Uk)) (5) where Ii: (.) is the density f the system nise at time k. Ik (Xk) is the psterir density functin at time k. It+l (Yk+ll xk+l) is the cnditinal likelihd density given by n.. (Yk+llxk+l) = Ii:+l (Yk+l - hk+l (Xk+l)) (6) where Ii: (.) is the density ffthe measurement nise at time k. The Kalman filter can slve the abve equatin analytically if the system is linear and the psterir density is kept as Gaussian. Fr mst f the ther nnlinear and nn Gaussian systems the analytic slutin t frmula (3) and (4) is intractable because the cmplexity f density functins increases ver time. One way t get the subptimal slutin is t apprximate the density functin by a sum fbase functins as (7) where ui is the weight f cmpnent i and f i is the parameter vectr that cntrls the shape r lcatin f cmpnents. With different frms f cmpnents and ways t generate them varius nnlinear filters are utilized. In this paper we chse three representative nnlinear filters fr evaluatin. In the fllwing the majr characteristics f these three filters are briefly intrduced. The detailed algrithms are beynd the scpe f this paper and can be fund in the references. (1) (2) A. Furier-based filter Furier series were first emplyed t estimate prbability densities in [14]. Recently [8] ensured the nn-negativity f Furier series by apprximating the square rt f the density instead f the density itself. The density functin f (x) is apprximated by its truncated Furier series expansin as T ensure nnnegativity the abslute square rt f Furier series is explited. In additin by Furier series the expressins in frmulas (3) and (4) can be calculated analytically while the type f density functin is preserved [8]. Because f the rthgnality f Furier expansins the cefficients f cmpnents can be calculated analytically. In practical implementatin it is dne by the Fast Furier Transfrm (FFT) algrithm. Additinally the number f cmpnents i.e. the cmplexity f calculatin can be adjusted ptimally. Because the imprtances f cmpnents are srted directly by the abslute value fcefficients with respect t a density distance measure cmpnent reductin is very efficient. Hwever the Furier transfrmatin by FFT usually requires a fine sampling reslutin in the whle state space. This results in high cmputatinal effrt and memry cnsumptin especially fr high-dimensinal density functins. B. Particle filter Particle filters represent the density by a series f randmly selected particles. The well-knwn Sequential Imprtance Sampling (SIR) particle filter cnsists f the fllwing steps. 1. Generate a new particle fr each ld particle randmly using the mtin mdel. 2. Weight the new particle using the measurement mdel i.e. likelihd densities. 3. Resample the particles delete the particles with small weight and split the particles with large weights. Finally each particle has the same weight. The advantage f the particle filter is that it is easy t be implemented and it simplifies the density integrals. With the increasing number f particles the filter can als handle arbitrary density functins. Hwever althugh particle filters can beat the curse f dimensinality in terms f cnvergence [11] they d nt beat the curse with respect t cmputatinal cmplexity [12]. Furthermre the rate f cnvergence is nt unifrm fr dynamic systems [13]. This implies that it is difficult t evaluate the quality i.e. the number f needed samples f such estimatrs. Additinally since particle filters are randm appraches the final result varies especially when the number f particles is small. C. Grid-based filter I (x) = L wiejitre. (8) In grid-based methd the density is sampled at unifrmly distributed grid pints. The integral in Bayesian estimatin equatin is replaced by a discrete summatin. Like the ther tw filters grid-based filter can als apprximate arbitrary density functins with increasing reslutin f grid pints.

3 But like the Furier filter the grid reslutin is the same in the whle state space. This leads t a high cmputatinal effrt and memry cnsumptin if a fine sampling r high dimensinality is required. Figure (1) illustrates the apprximatin f a Gaussian mixture density with three cmpnents by a Furier-based filter a particle filter and a grid-based filter respectively. Theretically all these three filters cnverge t the ptimal Bayesian slutin when the number f cmpnents increase twards infinity. But in practical systems it is very imprtant t chse a prper filter that has a fast cnverging rate t get an accurate result with a shrt cmputatinal time. Tw factrs affect the cmputatinal efficiency. One is the number f cmpnents that can accurately apprximate the density functin. This factr depends highly n the characteristics f specific system. The ther ne is the cmputatin time given different number f cmpnents. Table I lists hw the cmputatin time f each step increases with the number f cmpnents fr the abve three filters. We see bviusly that particle filter tends t be mst efficient when the number f cmpnents is large. Furthermre we can ntice that the efficiency f Furier-based filter is bth affected by the number f cmpnents f FFT sampling and the reduced number f cmpnents. III. NONLINEAR FILTERING IN INDOOR WLAN POSITIONING The indr psitining system we are wrking n is based n the received signal strength (RSS) f WLAN system. In a typical scenari a number f statinary access pints (APs) are distributed in an indr envirnment. Mbile devices measure the RSS signals which are used t estimate the psitin f mbile devices. In mathematic language the RSS signals are nted as p = [pi p2... pnj where pi is the RSS measurement frm AP i and the psitin f mbile device is nted as x = [x y z]. The functin h k (.) is used t describe the relatinship between the psitin and the RSS at time k as (9) where v k is the additive nise caused by the cmplicated indr envirnment. Table I COMPUTATIONAL COMPLEXITY ANALYSIS Furier-based Filter Transitin density apprximatin by m cmpnents (FFT) Reduce t n cmpnents Predictin Likelihd density apprximatin by m cmpnents (FFT) Reduce t n cmpnents Update Particle Filter Predictin Update Resampling Grid-based Filter Predictin Update Tn) (nlgn) (n) (n) Tn) O(n 2 ) (n) The first prblem is hw t mdel the mapping functin hk (.). This functin may be determined using a parametric radi prpagatin mdel e.g. multi-wall mdel r dminant path mdel [3]. But the perfrmance is nt always satisfactry due t the fact that the radi prpagatin is highly affected by many field-specific parameters such as walls drs and s n. These parameters are smetimes difficult t retrieve within the mdels r even nt knwn t the user. Fig. 2 shws the building map f ur test envirnment and a radi distributin map in this envirnment. We can see that the building structure has a clear influence t the radi distributin. Additinally RSS nise has als a cmplex frm. As Fig. 3 shws RSS nises at reference pints have varius and cmplicated frms which can hardly be mdeled by a simple Gaussian. In practical systems peple usually mdel the mapping functin and nise in a nn-parametric way i.e. measuring the RSS distributin ffline r nline at reference pints t build a nnparametric radi map r mapping functin. Anther prblem is hw t mdel the mtin f mbile devices. Unlike the car navigatin system the indr mtin is usually hard t predict because the mtin directin and distance vary largely. With the assistance f extra sensrs like accelermeter r gyrscpe the mtin behaviur can be Apprximatin Based Apprximatin Based Apprximatin Based n Furier Series n Randm Sampling n Grid Sampling Figure 1. Illustratin f density estimatin with 3 cmpnents X(m) 45 Figure 2. A radi map example in an ffice building.

4 016[ u. OOO[ Ol ::1 Figure 4. Left: simulated envirnment with 6 APs. Right: radi map example by multi-wall mdel RSS (dbm) Figure 3. Examples f RSS distributin at reference pints. 5 u mdeled parameterized. In this paper we assume the distance is measurable by either accelermeter r dmeter []. But the angle is unknwn. Hence the fllwing frmulas are derived t mdel the system: Xk+l] [ Yk+l [Xk] [(d k+<5dk ) COS((}k) ] Yk + (dk+<5dk ) sin (Ok) v» = hk ([ :: ]) + Vk+l (11) where distance nise <5d k fllws Gaussian distributin unifrm distributed variable ()k- The bservatins P k is the RSS vectr. The bservatin nise Vk+l is nn-gaussian. EXPERIMENTAL RESULTS The test envirnment in this paper is ur ffice building which has several WLAN access pints installed. T better evaluate and understand the perfrmance f different nnlinear filters we n1ake bth simulatins and field tests. In the simulatin a big area and dense radi map is generated and the mdel parameters are perfectly knwn. But in the field test nt t many reference pints are used t cnstruct radi maps and the mtin angle des nt fllw the presumed unifrm distributin. In what fllws the setups f simulatin and field test are described and results are then discussed. A. Descriptin f the Simulatin We started with simulated data because the walking trace in simulatin is generated frm the knwn mdel parameters which eliminates influence f the mdel errr in the cmparisn. We simulate the radi distributin in ur ffice building by a site-specific multi-wall radi prpagatin mdel as P = P -nlg(d) - L WAF + v Figure 5. density. 0 RSS nise (db) 20 Left: Gaussian-mixture nise mdel. Right: simulated transitin () N (0 a). The unknwn mving directin is mdeled by an IV. l (12) where P stands fr the AP transmissin pwer in dbm which is set t -20 dbm. n is the radi attenuatin cefficient which equals t W AF represents wall attenuatin factr that is the partitin value f walls between transmitter and receiver. Here we set WAF t 8.7 db fr cncrete walls 5.5 db fr glass windws and 4.3 db fr drs. 6 access pints are assumed t be available in an area f 45m x 45m as the left figure f Fig. 4. The RSS nise v is a tw-cmpnents Gaussian mixture density 0.5N (-7.522) + 0.5N (7.522) as shwn in the left figure f Fig. 5. Nte that althugh we use a parametric mdel t generate radi maps it is still stred in a nn-parametric way i.g. the RSS vectrs n grid pints are stred in cmputer fr the calculatin f likelihd functins. The simulated walking trace fllws the frmula (). All the mdel parameters are knwn t the estimatr. The distance between tw samples is assumed t be a Gaussian variable N (2.5m 0.3m 2 ). The mtin angle is unifrmly distributed between -7r and n S the transitin density fr each psitin is actually a ring as illustrated by the right figure f Fig. 5. Since we assume that the distance is the same fr each step the transitin density is als cnstant. At the update step a discrete likelihd prbability is made by calculating the likelihd given the measured RSS at each pint f the radi map. B. Descriptin f Field Test We als evaluate the filters using the real data. In ur building we installed 14 access pints as shwn in Fig. 6. We measured the radi distributin at 250 unifrmly distributed grid pints in an area f 15m x 35m befre the system runs. The RSS distributin at each pint is different because f the cmplex indr envirnment. Fig. 3 shws tw examples f RSS distributins. These distributin functins are stred in cmputer as a vectr. In the nline step RSS and accelermeter data are cllected while peple mve in a path as Fig. 6. We als use frmula () and (11) t mdel the system. But the distance dk is estimated by accelermeter and varies i.e. we must calculate the transitin density at each recursive step. The likelihd density is calculated in the same way as in the simulatin. At each grid pint the RSS measurement is cmpared with the stred RSS prbability density functin t

5 0.06 u 0.04 I u Figure 6. Envirnment f field test. get the likelihd value. Finally a discrete likelihd density functin is cnstructed fr updating. Q.O.Q1 Figure 8. Examples f psterir density functins in the field test. C. Results and Discussins Fig. 7 and Fig. 8 shw true psterir density functins at fur successive steps in bth simulatin and field test. These densities are generated by the grid-based numerical methd. We can bserve that the psterir density shws a strng nnlinearity which favrs the usage f nnlinear filters. T cmpare the perfrmance f different filters we take the mean f lcatin errrs and cmputatin time per step as evaluatin criteria. Fig. 9 and Fig. indicate hw the mean f lcatin errrs is like given different number f cmpnents. Nte that fr Furier filter we take 000 pints in simulatin and 750 pints in the field test fr FFT sampling. It is bvius that all the abve nnlinear filters cnverge almst t the same value given enugh cmpnents. Furier filter cnverges with fewest cmpnents. Grid-based filter is wrst. This indicates that the samples f Furier-based filter is mre efficient t represent the density functins. In additin the results f the particle filter are the averaged ne ver 25 runs. Since particle filter is a randm apprach its result als varies. Fig. 11 shws hw the result f particle filer varies in the field test given different number f cmpnents. It is clearly bserved that the Figure 7. Examples f psterir density functins in simulatin. :[6 Lb5 c: 4.. 3: m2-_.3. '+- c:. 411t I 0 NumberfCmpnen Figure 9. Mean f lcalizatin errrs given different number f cmpnents fr simulated data fewer particles lead t a larger variatin. Fig. 12 and Fig. 13 cmpare the cmputatin time given different number f cmpnents. All the cmputatins are dne in a laptp with a single-cre 2 GHz CPU. As shwn fr grid-based filter the cmputatin time increases quadratically while fr particle filter the cmputatin time increases linearly. In simulatin we use much mre grid pints than the field test t cnstruct radi map which means calculating the particle weight takes mre time in simulatin than in the field test because every particle must find the clsest grid frm all f them. That is why it takes mre time in simulatin than in the field test given the same number f particles. Furier filter has a almst cnstant cmputatinal time. This can be explained by the fllwing facts. As shwn in Table I Furier filter cnsists f tw parts f calculatins. The cmplexity f Furier transfrm part depends n FFT sampling fllwing the cmplexity f (m lg m). The ther part depends n the reduced number f cmpnents fllwing the cmplexity f 0 (n 2 ). Hwever if the number f Furier cmpnent is already sufficient t represent the densities adding mre cmpnents is just adding

6 Number f Cmpnents 0 (i) 4 en 3 c. 2 0) c " 1 ::J 5 ll.-i ---' ' ' J Number f Cmpnents Figure. Mean flcalizatin errrs given different number f cmpnents fr real data. Figure 12. Cmputatin time per step given different number f cmpnents fr simulated data. -: 2.5 I 14 : : en - Mean f 25 Runs 5 12 c Maximum f 25 Runs en Minimum f 25 Runs " 8. E :' -; m 1 0 0) 0 6 c...j "c; '0 4 s 0.5 c: c Number f Cmpnents Number f Cmpnents Figure 11. Variatin f mean lcalizatin errrs given different number f cmpnents fr real data by particle filter. Figure 13. Cmputatin time per step given different number f cmpnents fr real data. sme zer Furier series and nt increasing the running time. S as in Fig. 12 and Fig. 13 the FFT sampling m is fixed meaning a cnstant running time which als dminates the calculatin when the number f reduced cmpnent n is small. Since the Furier filter can apprximate the densities accurately with very few cmpnents (arund 200 as shwn in Fig. 9 and Fig. ) mre cmpnents will nt increase the cmputatin time neither. That is why the running time f Furier filter in Fig. 12 and Fig. 13 lks like cnstant. We als remark the imprtance f FFT sampling in Furier filter by Fig. 14 and Fig. 15 which shw hw the accuracy and running time change given different FFT sampling reslutins and a fixed reduced number f cmpnents. We can ntice that mre accurate FFT sampling gives better result but takes mre time. Fig. 16 and Fig. 17 plt the running time versus the mean f lcalizatin errrs which helps us t identify which filter makes the best trade-ff between efficiency and accuracy. In ur simulatin Furier-based filter and particle filter have a cmpatible perfrmance. But in real test particle filter perfrms best. This is because the density functin in simulatin is mre cmplicated and in simulatin the transitin density is the same fr all steps s sme cmputatin time can be saved by reusing the previus transitin density fr Furier filter. V. CONCLUSIONS In this paper we cmpare the perfrmance f three nnlinear filters fr indr WLAN psitining prblems. We use bth simulated and real measured data fr evaluatin. Frm the results we see that given enugh cmpnents all three filters cnverge t the same result. Given very few cmpnents Furier-based filter perfrms best in the sense f accuracy e.g. less than 200 cmpnents can already give very accurate results in ur cases. The trade-ff number depends n the cmplexity f system. Particle filter is faster than thers fr its simple algrithm structure with linear cmplexity. Furier is slwer with few cmpnents because the Furier transfrm and cmpnent reductins take sme extra time. This time is clse t be cnstant i.e. nt increasing with mre cmpnents which can be regarded as fixed verhead cst. With mre cmpnents time cnsumptin f grid-based methd is clse t 0 (n 2 ) while that fparticle filter is clse t (n). Fr Furier filter the cmputatin time increases with the density cmplexity instead f the number f cmpnents i.e. if the current cmpnents are enugh t represent the density the mre added cmpnents are zers taking very little time in calculatin. Therefre it is almst cnstant even using large number f cmpnents. The sampling reslutin f Furier filter is imprtant. The larger reslutin is used the

7 1 I f!?4 e W j ' il-- _ Reslutin f FFT Sampling 1500 Figure 14. Mean f lcalizatin errrs given different FFT sampling reslutin and a reduced 200 cmpnents. 3.5 : : 3 l2.5 CD 2 E 1.5 0) c ' 1 0:: GF -PF - - -FF Mean f Lcalizatin Errrs (m) Figure 16. Running time per step given mean f lcalizatin errrs fr simulated data. 0.3 : :0.25 c. s 0.2 a. E0.15 F 0.1 'c c: & FF - GF -PF Reslutin f FFT Sampling Mean f Lcalizatin Errrs (m) Figure 15. Running time per step given different FFT sampling reslutin and a reduced 200 cmpnents. Figure 17. data. Running time per step given mean f lcalizatin errrs fr real mre accurate result is achieved. But als mre time is spent which slws the Furier filter in high-dimensinal prblems. Fr practical psitining systems with limited cmputatinal pwer e.g. slw prcessrs particle filter is the best chice. If the density functins are transfered between different ndes Furier-based filter shuld be used because fewer cmpnents require smaller bandwidth and cmmunicatin time. REFERENCES [1] Bahl P and Padmanabhan VN "RADAR: An in-building RF-based user lcatin and tracking system" in IEEE INFOCOM 2000 pp [2] Rs T Myllymaki P Tirri H Misikangas P and Sievanen J "A prbabilistic apprach t WLAN user lcatin estimatin" Internatinal Jurnal f Wireless Infrmatin Netwrks vl. 9 n. 3 July [3] Henning Lenz Brun Betni Pardi Hui Wang Andrei Szab Jachim Bamberger Jachim Hrn and Uwe D. Hanebeck "Adaptive Lcalizatin in Adaptive Netwrks" Chapter f Signal Prcessing Techniques fr Knwledge Extractin and Infrmatin Fusin. Springer [4] R. E. Kalman "A new apprach t linear filtering and predictin prblems" Transactins fthe ASME-Jurnal fbasic Engineering vl. 82 (Series D) pp [5] Brank Ristic Sanjeev Arulamplalm Neil Grdn Beynd the Kalman filter. Artech Huse Bstn [6] Julier Simn 1. and Jeffery K. Uhlmann. "A New Extensin f the Kalman Filter t nnlinear Systems." in The 11th Internatinal Sympsium n Aerspace/Defense Sensing Simulatin and Cntrls Multi Sensr Fusin Tracking and Resurce Management II SPIE [7] Marc Huber Dietrich Brunn and Uwe D. Hanebeck "Clsed-Frm Predictin f Nnlinear DYnamic Systems by Means f Gaussian Mixture Apprximatin f the Transitin Density" in 2006 IEEE Internatinal Cnference n Multisensr Fusin and Integratin fr Intelligent Systems (MFI2006) pp [8] Dietrich Brunn Felix Saw and Uwe D. Hanebeck "Efficient nnlinear Bayesian estimatin based n Furier densities" in 2006 IEEE Internatinal Cnference n Multisensr Fusin and Integratin fr Intelligent Systems (MFI 2006) pp [9] Oliver C. Schrempf Dietrich Brunn and Uwe D. Hanebeck "Density Apprximatin Based n Dirac Mixtures with Regard t Nnlinear Estimatin and Filtering" in 2006 IEEE Cnference n Decisin and Cntrl (CDC 2006) Dec [] Hui Wang Henning Lenz Andrei Szab Jachim Bamberger and Uwe D. Hanebeck "WLAN-Based Pedestrian Tracking Using Particle Filters and Lw-Cst MEMS Sensrs" in Wrkshp n Psitining Navigatin and Cmmunicatin (WPNC 2007) Mar [11] Arnaud Ducet Simn Gdsill and Christphe Andrieu "On Sequential Mnte Carl Sampling Methds fr Bayesian Filtering" Statistics and Cmputing vl. n. 3 pp [12] Fred Daum and Jim Huang "Curse f Dimensinality and Particle Filters" Aerspace Cnference Prceedings IEEE March [13] Mirslav Simandl and Ondej Straka "Sampling densities f particle filter: a survey and cmparisn" American Cntrl Cnference (ACC '07) pp [14] R. Krnmal and M. Tarter "The Estimatin f Prbability Densities and Cumulatives by Furier Series Methds" Jurnal fthe American Statistical Assciatin vl. 63 n. 323 pp September 1968.