Time of Arrival Estimation for WLAN Indoor Positioning Systems using Matrix Pencil Super Resolution Algorithm

Transcription

1 Tim of Arrival Estimation for WLAN Indoor Positioning Systms using Matrix Pncil Supr Rsolution Algorithm Ali AASSIE ALI 1, and A. S. OMAR 2 FEIT- IESK,Chair of Microwav and Communication Enginring Postfach 4120, Univrsity of Magdburg, Magdburg, Grmany 1 -mail: ali@is.t.uni-magdburg.d, 2 -mail: a.omar@i.org Abstract - Th accuracy of prdiction of tim of arrival (TOA) in wirlss local ara ntwors (WLAN) is th most important paramtr for indoor positioning systms. This papr prsnts th application of supr rsolution matrix pncil (MP) algorithm for TOA stimation for indoor positioning application. Also it prsnts th rsults of frquncy swp masurmnts of indoor channl for th WLAN IEEE standards (a, b, and g ). Th simulation and xprimntal rsults show that MP can accuratly stimat TOA and has suprior prformanc ovr th Fourir transform tchniqu. Also th computational complxity of th proposd tchniqu is compard with th ESPRIT and root-music algorithms. 1 Introduction Th positioning systms that ar usd to trac and dtrmin th usrs hav gaind incrasing intrst. A Global Positioning Systm (GPS) is a typical xampl which dpnds on th rcivd signal from multipl satllits. In indoor location systms GPS is not fficint du to obstruction and shilding of satllit signals. Also thr ar cllular ntwors bas wid ara location systms, which ar limitd by th cll siz. In indoor nvironmnt, thr ar many positioning systms basd on diffrnt tchnologis such as ultrasound [1], Infrard [2], vido survillanc and th systms that dpnd on rcivd signal strngth [3]. For indoor nvironmnt indoor Wirlss Positioning Systms (WPS) basd on WLAN infrastructur hav gaind grat attntion in rcnt yars [4][5][6]. A WLAN-basd positioning systm has advantags ovr all othr indoor positioning systms bcaus it is a part of th xisting communication structur and can covr many buildings. Th charactristics of radio signal in mobil nvironmnts ar utilizd to prdict th location of th mobil unit. Th basic charactristics ar Rcivd Signal Strngth (RSS), Angl of Arrival (AOA) stimation and tim of arrival TOA stimation. Th disadvantag of th RSS mthod is th random dviation from man rcivd signal strngth causd by shadowing and small scal channl ffct [7]. Th AOA rquirs antnna arrays at ach nod which incras th complxity of th xisting systm, as wll as, prforming wors in multipath nvironmnt. Th most important paramtr for accurat indoor positioning systms is th tim of arrival TOA of th Dirct Lin of Sight DLOS path [8]. In this cas accurat stimation of TOA from rcivd communication signals ar rquird. Indoor multipath intrfrnc is th main factor that limits dploying indoor positioning systms, th multipath is svr and complx which lads to inaccurat stimat of th TOA using convntional tchniqus. Also in systms that hav fixd bandwidth it is important to find altrnativ TOA stimation tchniqus. Diffrnt supr rsolution tchniqus such as Estimation of Signal Paramtrs via Rotational Invarianc 11

2 Tchniqus (ESPRIT) and Multipl Signal Classification (MUSIC) hav bn usd for spctral stimation applications. Supr rsolution tchniqus ar studid in th fild of TOA stimation in indoor positioning systms such as multipl signal classification MUSIC [8] and Root-MUSIC [9]. Supr rsolution tchniqus can incras tim domain rsolution but in this cas, th complxity of th systm implmntation also incrass. For xampl MUSIC hav a limitation which includs th computational complxity of covarianc matrix stimation. Th goal of this papr is to fficintly stimat TOA of a dirct ray with lowr complxity using supr rsolution matrix pncil MP algorithm [10] basd on frquncy domain masurmnts of indoor channl. This papr is organizd as follows: Sction 2 prsnts th indoor channl modl and th application of th MP algorithm for TOA stimation. Sction 3 dmonstrats th simulation rsults. Sction 4 dmonstrats th xprimntal masurmnt systm and th prdiction rsults and Sction 5 concluds th papr. 2 Multipath Indoor Radio Channl Th complicatd indoor radio propagation can b charactrizd using th basband complx impuls rspons which can b xprssd as: h(t ) = L = 1 a δ (t τ ) (1) j whr L is th total numbr of th dlayd paths, a = a θ is th complx amplitud, τ is th propagation dlay, and δ is th Dirac dlta function. Th paramtric modl of discrt complx frquncy domain indoor wirlss channl can b xprssd as: H( f n L ) = a = 1 j( 2πf n τ ) for n = 0,1,2,...N- 1 whr f n = fo +n f s, fo is th starting frquncy in Hz, f s is th frquncy spacing and N is th numbr of masurd points. Th frquncy domain transfr function can b writtn as: L H ( n ) n = A z (3) = 1 j2πf Whr A =a o τ j2πf, and z = s τ ar th pols in th Z plan. For th prdiction problm, it is rasonabl to assum that within a tim fram th scattring gomtry and multipath paramtrs ar tim invariant [11]. Th problm of finding multipath paramtrs and TOA can b rducd in a form that can b solvd using th MP algorithm. Fig. 1 shows th flowchart of th algorithm. At first th matrix H (N-M)x(M+1) is formd using th complx data squnc H(n) as follows: (2) H = H( H(0 ) H(1) N M 1) H(1) H( 2 ) H( N M ) H( M ) H( M + 1) H( N 1) (4) 12

3 whr M is calld th pncil paramtr. With larg valu of M th computation of high rsolution MP algorithm incrass. Th valu of M is to b slctd as a compromisd btwn rsolution and computation complxity. M is chosn mpirically btwn N/3 and 2N/3 to gt a good prformanc[10]. Aftrward th two (N-M)xM matrics H 1 and H 2 ar dfind as: H 1 = H( H(0 ) H(1) N M 1) H(1) H( 2 ) H( N M ) H( M H( N 1) H( M ) 2 ) (5) and H 2 = H( 1) H( 2 ) H( N M ) H( 2 ) H( 3 ) H( N M + 1) H( M ) H( M + 1) H( N 1) (6) By considring th following pncil matrix : H 1 -λh 2 (7) Which rprsnt a gnralizd ignvalu problm, th pols z can b calculatd from th principl ignvalus of (7). Th tim dlays can b stimatd as: Im( z ) τ = (8) 2πf s Th complx amplituds a of th impuls rspons can b obtaind by solving th linar systm of (2) which can b formulatd as a matrix quation: Whr H = SA (9) S = jω τ o 1 jω τ jω 1 1 τ N 1 1 jω τ jω τ jω o τ N 1 2 jω τ jω τ jω o L 1 L τ N 1 L 13

4 H [ H( ω ) H( ω ) H( ] T = ω o 1 N 1 ) A = [ a a ] T 1 2 al Whr ω=2πf n. Eq.(9) is an ovr dtrmind systm which can b solvd to find complx amplituds a i using linar last squars and psudoivrs approach as follow : whr * is th complx Hrmitian transpos. * 1 * A = ( S S ) S H (10) Estimation of Transfr function H(f) H Matrix formation Estimat L Finding ign valus of Matrix pncil Estimation of TOA, A i Fig.1 Rcivr of Supr rsolution MP TOA stimation systm MP algorithm nds to idntify and stimat th numbr of multipath componnts L to wor corrctly. Thr ar numbr of mthods basd on statistical classification critria, which can b usd to stimat L. In this papr, th Minimum Dscriptiv Lngth (MDL) and Aia Information Critria (AIC) ar usd[12] as follow: and MDL( 1 ) = log( f ( Θ )) + ( 2N )log M (11) 2 AIC( ) = log( f ( Θ )) + 2( 2N )M (12) whr f(θ) is th lilihood function and it bcoms th ratio of th gomtric man to arithmtic man of a numbr of th ignvalus. In th abov quations th valu of that minimizs th MDL and AIC rprsnts th numbr of multipath componnts L. 3 Simulation Rsults To dmonstrat th prformanc of MP algorithm in TOA stimation for WLAN indoor positioning application, xtnsiv simulations wr prformd. Comparison of th proposd algorithm with th convntional invrs Fourir transform (IFT) is prsntd. Th complx frquncy rspons was calculatd by simulating th indoor WLAN channl. Fig.2 shows th normalizd tim domain channl impuls rsponss obtaind from frquncy swpt data in th rang 2.4 to 2.48 GHz (IEEE801.11b WLAN standard). In this situation which appar in limitd bandwidth systms mas it impossibl to distinguish th DLOS path using th convntional IFT tchniqu. It shows th ability of MP for dtcting DLOS path, which corrspond to th TOA. 14

5 1.5 IFT MP Normalizd Amplitud Dirct LOS path First dtctd path using IFT Tim dlay (ns) Fig.2 Normalizd impuls rspons Fig.2 prsnts th rror prdiction prformanc as a function of th numbr of points of th complx frquncy rspons. W can obsrv that th rror dcrass as th numbr of points incras (high sampling frquncy). In convntional tchniqus in ordr to sparat two componnts of τ 1 and τ 2 with τ 2 > τ 1 it is ncssary to hav a frquncy rspons with a bandwidth of 1/( τ 1 - τ 2 ) % Error No of points Fig. 3 TOA stimation rror prformanc Fig. 4 dmonstrats th computational tim prformanc of th MP algorithm as compard to ESPRIT and root-music. It illustrats th advantag of using MP for TOA stimation as compard with root-music, and also shows that MP hav computation advantag compard with ESPRIT which ma MP algorithm mor attractiv for ral tim applications. 15

6 1 Normalizd computational tim MP ESPRIT MUSIC Fig.4 Computational tim prformanc comparison 4 Masurmnt Systm and Exprimntal Rsults On of th most popular tchniqus to xprimntally calculat th TOA is through th us of a frquncy domain masurmnt systm using Vctor Ntwor Analyzr (VNA). Th main componnt of th masurmnt systm usd is an Anritsu-Wiltron 37347A ntwor analyzr. Th complx frquncy channl rspons can b obtaind by swping th channl at uniformly spacd frquncis. Th forward transmission scattring cofficint S 21 (th complx channl frquncy rspons) is masurd. Th systm is dpictd in th bloc diagram of Fig.5. Tx Wirlss Channl Rx VNA Control, Signal procssing GPIB Fig. 5 Bloc diagram of VNA frquncy rspons masurmnt systm Th magnitud and phas of th masurd frquncy rspons wr stord for ach masurmnt and latr usd for furthr procssing. Th masurmnt systm was calibratd to rduc th ffcts of th masuring quipmnt. Also, th slf vctor rror corrction facility in th VNA allows th tracing rrors of th ntir masurmnt systm to b rmovd. All masurmnts ar automatd using PC with control through a Hwltt Pacard s vrsion of a Gnral-Purpos Instrumntation Bus (GPIB). Th frquncy rsponss ar collctd at 3 16

7 frquncy bands GHz, GHZ, and GHz. Thr ar 1601 complx data points for ach rspons. Th masurmnts ar prformd in th 3rd floor of th building of institut of lctronics, signal procssing and communication, Univrsity of Magdburg. Th layout of th masurmnt hall is shown in Fig.6. Fig. 6 Floor plan of th masurmnts location Th tim synchronization btwn transmittr and rcivr for TOA stimation is vry important for positioning systm. Th VNA systm oprats with a transmittd rfrnc (Tx- Rx synchronization) and allows th phas to b stabl. Using VNA for masurmnts will solv th problm of synchronization btwn Tx and Rx. Th calibration procdur sts th tim rfrnc points from th analyzr ports to th calibration points which, in our study, ar at th nd of th cabls. Whn th tim rfrnc is shiftd to th nd of th cabls (to th antnna connctors), th rsulting dlay profils includ only th propagation dlays that ar coming from th radio channl. Schmatic diagram of th synchronization procss of th VNA is shown in Fig. 7. RF sourc Wirlss Channl X IF filtr, Dtctor Rfrnc Channl X Frquncy Control Local Oscilator Fig. 7 Schmatic diagram of th transmittd rfrnc VNA Th masurmnts in th thr frquncy bands wr don with sparation distanc from 5 to 12 m btwn th transmittr Tx and th rcivr Rx in 1m stp for ach masurmnt. Th 17

8 Tx and Rx antnnas wr two idntical TEM horn antnnas. To rlat th stimation rsults of TOA to indoor positioning applications, tim dlays ar convrtd to distanc d btwn Tx and Rx as follow: d=c. min{ τ i i=1..l} (13) whr c is th spd of light in fr spac, 3x10 8 m/sc. A part from masurmnts, which corrsponds to frquncy rangs of WLAN standards as shown in Tabl.1 ar usd for th stimation of TOA and thn distancs btwn Tx and Rx. Standard Approvd IEEE a IEEE b IEEE g Frquncis of GHz, Opration GHz GHz GHz Tab.1 WLAN standards. Th MP applid to th frquncy channl rsponss thn th TOA and th distanc btwn Tx and Rx ar stimatd. Sampl from th masurmnts ar shown in Fig RMSE = Transfr function (db) Rconstructd Masurd Frquncy (GHz) Fig.8 Masurd and rconstructd frquncy rspons magnitud Th rsults of stimatd distancs btwn Tx and Rx (i.. TOA of dirct ray) ar shown in Fig.9. To masur th accuracy of th obtaind rsults th rspons is rconstructd from th stimatd paramtrs and th root man squar rror RMSE wr also calculatd. Rsults of rconstructions ar shown in Fig.8. 18

9 13 IEEE801.11a GHz IEEE801.11a GHz IEEE801.11b,g 11 Prdictd Distanc(m) Actual Distanc (m ) Fig.9 Rang prdiction prformanc Fig. 10 shows th valus of MDL and AKI as a function of th numbr of multipath componnts of masurmnts of Fig.8. Th minimum of MDL and AKI can b considrd as an stimat to th numbr of multipath componnts L MDL AIC Amplitud Conclusions Fig.10 MDL and AIC critrions as a function of th numbr of multipath componnts. In this papr, MP algorithm has bn applid to complx frquncy rspons to prform TOA stimation for indoor positioning application. Extnsiv masurmnts of frquncy rsponss for IEEE WLAN standards (a,b and g) hav bn don. Th simulation and xprimntal rsults outprform th high rsolution of th algorithm. Th rsults show that MP hav bttr prformanc than convntional IFT tchniqus. Also th computation prformanc of MP is found to b bttr than that of ESPRIT and MUSIC supr rsolution algorithms. 19

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