Small Range High Precision Positioning Algorithm Based on Improved Sinc Interpolation

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1 Small Range Hgh Precson Postonng Algorm Based on Improved Snc Interpolaton Zhengpng L, Chaolang Qn, Yongme Zhang, L a, and Changlu Nu Abstract Ths paper desgned an mproved postonng system whch employed a proposed mproved Snc nterpolaton algorm to reduce e samplng frequency of e system, calculated tme dfference of arrval (TODA) values w matched flter, estmated e poston of e target node by Chan's algorm and used Chauvenet crteron to optmze e postonng results. Analyss and smulaton results showed at e algorm could reduce e samplng frequency to a great extent whle ensurng e accuracy of e postonng system. Index Terms Wreless postonng, TDOA, Chan's algorm, Snc nterpolaton, Chauvenet crteron. I. INTRODUCTION Small range hgh precson postonng has broad applcaton prospects. It can be used for montorng and trackng ntensve care, preventng e baby stolen n delvery room and controllng e expensve medcal equpment n e medcal ndustry. Small range hgh precson postonng s urgently needed n nteractve games. The exstng postonng technologes cannot meet e requrements of e forementoned applcatons. The GPS satellte postonng accuracy can be very hgh, but when e GPS recevers are ndoor, e sgnal streng s greatly attenuated because of e buldngs and objects []. Some companes have developed small range postonng devces, for example, Xsens proposed VN otongrd for small range postonng and e accuracy can reach cm. But e employed postonng algorms aren't open for research. Tme dfference of arrval (TDOA) can be used for hgh precse postonng system, but few researches have been don't on how to employ TDOA to enhance e precson []. Ths paper desgned an mproved postonng system whch employed e proposed mproved Snc nterpolaton algorm to reduce e samplng frequency of e system for TDOA value. Then e system estmated e poston of e target node by Chan's algorm and used Chauvenet crteron to optmze e postonng results. As e result, we can reduce e samplng frequency to e maxmum extent on e premse of ensurng e accuracy. The rest of e paper s organzed as follows. Secton II made a survey of e related researches, Secton III descrbed e related e prncple of e proposed algorm, Secton IV made smulaton and analyss of e algorm, and anuscrpt receved December, 5; revsed arch, 6. The auors are w Nor Chna Unversty of Technology, Chna (e-mal: zhplee@gmal.com). Secton V drew a concluson. II. RELATED RESEARCH The concept of wreless locaton system has been proposed as early as 3 years ago. W e development of technology and e ncreasng demand of postonng, wreless locaton technology has made great progress [3]. At present, e postonng meod based on dstance measurement s manly based on e dstance estmaton between nodes to locate e unknown nodes. Dstance estmaton was done by measurng e dstance or angle of e nput sgnal. The exstng technques nclude: e receved sgnal streng (RSSI), tme of arrval (TOA), tme dfference of arrval (TDOA) etc. We manly talk about e TDOA technques. TDOA technology estmates e locaton of e target node by detectng e tme dfference of sgnal arrvng dfferences among reference nodes. The tme dfference of e same sgnal to e two reference nodes determnes e hyperbolc of a target node, e locaton of e target node two can be gotten w two hyperbolc tracks [4]. Accordng to e prncple of TDOA postonng, f we can accurately get e tme dfference of e arrval, we can calculate e exact locaton of e target node, so e accurate estmaton of sgnal arrval tme dfference of dfferent reference nodes s e key of e TDOA postonng. Based on e TDOA and e correlaton technologes, Zhengpng L and Zren Wang (3) put forward a new algorm at can greatly mprove e postonng accuracy. Ther system employs matched flter to calculate TDOA value and does not need precse synchronzaton between e transmtter and recevers so at t makes e TDOA value more accurate. Ther postonng accuracy can be less an cm [5]. The problem s at e postonng gets hgher accuracy when e samplng frequency s GHz. It s dffcult to realze n e actual system. Based on at, s paper proposed an mproved postonng system to reduce e samplng frequency n e premse of ensurng e postonng accuracy. III. RELATED ALGORITH AND THE POSITIONING PROCESS In e paper Precson Wreless Postonng Scheme n Small Range Based on Frst-Order Dfference and Correlaton Inspecton. Ther algorm process s as follows: In e recever, e system frst dd ampltude normalzaton processng of receved F wave. Then t dd match flterng between e receved sgnal after ampltude normalzaton processng and e match flter sgnal whch was smlar to e modulated sgnal of e transmtter. The do:.878/jfcc

2 result shows n Fg Fg.. atched flterng output sgnal. A matched flter nput sgnal was s (t), e mpactresponse was h( t) s( t t), e output of e matched flter was equal to e convoluton of e nput sgnal and e mpact response: s out x -7 ( t) s( t) h( t) s( ) h( t ) d kr ( t t) () Formula () showed at e output sgnal of matched flter and e autocorrelaton functon of lnear F sgnal were e same n form, just had dffers of a constant factor k and a tme delay t. The autocorrelaton functon, R S ( t t) 's maxmum s R (), so e output sgnal of matched flter get S t t. e maxmum when But er problem was at e matched flterng output sgnal gets a good result when e samplng frequency s GHz. It s dffcult to realze n e actual system. Based on at, we added e mproved Snc algorm before e matched flterng process to reduce e samplng frequency. From results of e matched flterng output sgnal w mproved Snc algorm shown n Fg. and Fg. 3, t could be seen at e peak of e matched flter result was also qute obvous. But compared to Fg., ere were multple peaks n e Fg. and Fg. 3. The multple peaks could have a certan degree of nfluence on e postonng accuracy but not great. In e four secton, system smulaton and analyss, we would see e concrete results s x -7 Fg.. atched flterng output sgnal w mproved Snc algorm-ghz. Because e receved sgnal had a tme delay compared w e matched flter sgnal, e output sgnal could be ganed by dong matched flterng between ese two sgnals. When e output got e maxmum, e tme was t. A group of dfferent t values could be gotten by dong matched flterng between e modulaton sgnal and e receved sgnal of each recever. Suppose at t of recever was T, e t of recever was, T T T. Then e TDOA value was. After gettng a group of TDOA values, e poston of target node could be estmated by Chan's algorm. The Chan's algorm was ntroduced n detal n e paper Precson Wreless Postonng Scheme n Small Range Based on Frst-Order Dfference and Correlaton Inspecton, so we needn't repeat e algorm here x -7 Fg. 3. atched flterng output sgnal w mproved Snc algorm-5hz. A. The Postonng Process Ths paper presented a hgh accuracy postonng meod n small range based on an mproved Snc nterpolaton algorm, e steps are as follows: Step : L fxed target nodes receved e F wave sgnals from e reference node, of whch L was a postve nteger. The modulaton sgnal of e F wave was sawtoo sgnal, and so e F wave was called sawtoo F wave. A cycle of e sawtoo F sgnal was called a chrp; Step : We conducted ampltude lmtaton on e receved sgnals, sampled on contnuous chrps at nterval T and acheved e sample functon x n whch =,,, ; n=,,,n. N was e sample pont number of each chrp. Step 3: By usng mproved Snc nterpolaton algorm to reconstruct e sample functon,we could get e reconstructon functon y (k) of whch k=,,, (N-)(+) +. Step 4: Samplng e orgnal sawtoo F waves at nterval T/(+) and achevng e sample functon u(k) where k=,,,(n-)(+)+. Then, we took u (k) and y (k) to perform e cross-correlaton operaton and obtaned e correlaton peak locaton A of whch =,,, L. Step 5: Usng correlaton peak locaton gap we could calculate e sgnal arrval tme dfference t t,, t t between e, 3, 4,, L,, 3,,, L, 7

3 reference node and e reference node. Among em t t,, t t were e TDOA values of whch,, 3,,, L, t ( A A ), * T. T was e tme nterval of e sample ponts n y (k) ; Step 6: The TDOA values and e coordnates of e reference nodes was put nto Chan algorm to calculate e poston of e target node. Step 7: Chauvenet crteron was used to optmze e postonng results. B. The Improved Snc Interpolaton Algorm In Step 3 an mproved Snc nterpolaton algorm was mentoned, and a detal descrpton of e mproved Snc nterpolaton algorm was present n s secton. We used e Snc nterpolaton algorm to reconstruct e sample functon x (=,,, ; n=,,, (N-)(+)+) obtaned from e Step,. ponts were nserted between e two neghborng samplng ponts and form a new sequence ( m, (, m=,,, ; n=,,, N-) from followng formula. c ( m, sn[( n m /( ) l) ] N x ( l) l ( n m /( ) l) We combne x and e ponts nserted nto e samplng ponts to obtan e reconstructed sequence y (h) : Namely: y () x (), y ( ) x (), y (( N )( )) x ( N ), y () c (,), y ( ) c (,), y (( N )( ) ) c (, N ), y ( ) c y ( ) c (, ),,, (, ),, y (( N )( ) ) c x ( y ( h) c ( m,, m,,, n,,,, N (, N );, h n( ), h n( ) h,,,,( N )( ) () (3) (4) y ( ), y ( ), y ( ),, y (h) were used to h h 3 h reconstruct agan and we obtaned e reconstruct sequence c y (k) : ) y (), ) y ( ),, ( N )( )) y (( N )( )), ) y (), ) y ( ), ( N )( ) ) y (( N )( ) ), ) y 3) y ( N )( ) ) y ) y ( ), ) y ( ),, ( N )( ) ) y (( N )( ) ) ) y (), ( ), ) y ( 3),,, ( ),, ( N )( ) ) y Namely: (( N )( ) ), (( N )( ) ); n( )), k n( ) k) y ( n( ) ), k n( ) n,, N,,,3,, Fg. 4. showed e reconstructed sequence k). () x c (,) c (,) c c (, (, Fg. 4. The reconstructed sequence k) w e mproved Snc nterpolaton algorm. C. Data Processng x () x x ( n ) In e results of postonng, multpa sgnal nterference would make some TDOA values greatly devatng from e dstance between e target node and e reference nodes. We called ese TDOA values e abnormal data. If e abnormal data was used to calculate e target node s poston, e postonng error would be very large. Therefore we should elmnate e abnormal data to ncrease e postonng error. In e paper Precson Wreless Postonng Scheme n Small Range Based on Frst-Order Dfference and Correlaton Inspecton, ey employ e frst-order dfference meod. The frst-order dfference meod s very sutable for e real tme data acquston and processng (5) (6) 7

4 system. But e precson of e frst-order dfference meod s also related w e measurement precson of e frst two values, except e sze of e error wndow. If e measurement s not monotonc ncreasng or decreasng functon, s meod wll produce large errors at e nflecton pont. Chauvenet crteron s also known as e equal probablty prncple, t s a very common crteron to elmnate error n actual project. In s system, n Step 7 we used e Chauvenet crteron to search for e values of large error and take em out. Chauvenet crteron was as follows: Durng e n measurement, we took e number at mght not occur for.5, so for e normal dstrbuton, e probablty at e error never appear was: mean square error (RSE) of postonng results, whch was frequently-used at present. The postonng system smulaton was done n dfferent condtons rough atlab: ) The postonng accuracy w dfferent Snc nterpolaton algorm was shown n Fg. 5. RSE/m RSE of dfferent algorms - Samplng Frequence 5Hz non-mproved Snc nterpolaton algorm mproved Snc nterpolaton algorm n n x exp( ) dx (7) n Accordng to e known value n located n e rght sde of e equaton, we could use e table of e standard normal functon to fnd out e Chauvenet coeffcent For e n measured value x, f e absolute value of e dfference between e measured value and e average was greater an e product between e standard devaton and Chauvenet coeffcent, e measured value should be elmnated. That was x x nsx, where Sx was e standard devaton Carrer Frequency/Hz Fg. 5. RSE of dfferent Snc nterpolaton algorm. TABLE I: RSE OF DIFFERENT SINC INTERPOLATION ALGORITH RSE/m CF/Hz RSE square RSE crcle RSE astersk IV. SYSTE SIULATION AND ANALYSIS In e smulaton system, e postonng area was determned by e number of reference nodes, and e more number of reference nodes e larger of postonng area. The reference nodes were statonary and ey should be dstrbuted around e postonng area unformly as much as possble so at e system could get better postonng results. If e reference nodes were changed to dstrbute at random or n a straght lne n e postonng area, e postonng result would not be as well as before. In s smulaton system, t supposes at e postonng range s m m. The coordnates of seven reference nodes were (,), (,), (,-4), (,), (,4), (,), (-4,). Target node was act as transmtter. The modulaton sgnal s frequency of target node was Hz. The smulaton supposes at e transmsson channel was 6 pas Rcan channel whch had one lne-of-sght (LOS) pa and 5 reflecton pas. The reflecton pas were caused by e multpa effect because of e sgnal reflecton, dffracton and scatterng. The addtonal delay of 6 pas were [ 3/e9 7/e9 9/e9 73/e9 5/e9] (s); e addtonal attenuaton were [ ] (db) and s was a common ndoor channel. In addton, e receved sgnal summed togeer of e LOS sgnal and reflecton sgnals. If e obstacles were on e LOS pa, t sould affect e TDOA value and cause TDOA errors. The postonng accuracy was measured w e root In Fg. 5, abscssa was carrer frequency and e values were Hz, Hz, 3Hz, 4Hz,5Hz, 6Hz, 7Hz,8Hz, 9Hz and Hz. Ordnate was RSE. Table I lsts e detals of each pont n e Fg. 5.The non-mproved Snc nterpolaton algorm referred at reference node receves each chrp and en nterpolate ponts behnd every samplng ponts of e samplng chrp [6]. We assumed every samplng chrp contans pont and e nterpolaton ponts are 9. So e nterpolated chrp contaned ponts. Fnally we used e nterpolated chrp to do correlaton detecton w e frequency-modulated wave at e target node transmtted and retreved e poston of e correlaton peak to calculate e TDOA values. The mproved Snc nterpolaton algorm referred at e reference node receved a chrp and en nterpolate ponts behnd every samplng ponts of e receved chrp. But s tme we took out some ponts from 9 successve chrps to form a new chrp to do e correlaton detecton. The pont takng out rules: We regarded e samplng chrp (totally ponts) as e bass. Because we had nterpolated 9 ponts behnd every 73

5 samplng pont, every nterpolated chrp has ponts. We took out e nd,,,, 999 nterpolaton st rd pont of e chrp; e 3, 3, 3,, 9993 rd nterpolaton pont of e chrp; ;e,,, nterpolaton pont of e 9 chrp. We totally take out 9 ponts and en we nterpolated e 9 pont to e bass (namely e chrp) to form e new chrp to do e correlaton detecton. The samplng frequency of ree algorms n e Fg. was 5Hz. From e smulaton results t could be seen at when we used e non-mproved Snc nterpolaton algorm, e postonng accuracy had not been sgnfcantly mproved. The RSE only decreased from about 3.8m to about 3.4m. But e mproved algorm, comparng w e non-mproved algorm n postonng accuracy, had a very large enhancement. The RSE only decreases from about 3.8m to about.m. ) The postonng accuracy n dfferent samplng frequency and nterpolaton ponts were shown n Fg. 6: RSE of dfferent algorms Table II lsts e detals of each pont n e Fg. 6. The target node was statonary and t could be at anyplace n e postonng area. From e smulaton results t could be seen at e algorm w mproved Snc nterpolaton comparng w algorm wout mproved Snc nterpolaton n postonng accuracy had a very large enhancement. When e samplng frequency was reduced from GHz to GHz and 5 Hz wout mproved Snc nterpolaton algorm, postonng accuracy was obvously declned. When e samplng frequency was GHz, e RSE value was qute low but e samplng frequency was GHz whch was too hgh to acheve n practcal applcaton. So we reduced e samplng frequency to GHz/5Hz and used e mproved Snc nterpolaton algorm to nterpolate 9/9 ponts to e samplng chrps as far as possble to acheve e GHz samplng frequency accuracy. The results showed at e accuracy of e samplng frequency GHz/5Hz w nterpolaton pont 9/9 was very close to e samplng frequency GHz. We could also see from results at e carrer frequency had lttle nfluence on e postonng accuracy when e carrer frequency varyng from Hz to Hz. 3) The postonng accuracy n dfferent numbers of nterpolaton ponts were shown n Fg. 7: RSE of dfferent nterpolaton ponts RSE/m - Samplng Frequence 5Hz Samplng Frequence 5Hz w nterpolaton ponts 9 Samplng Frequence GHz Samplng Frequence GHz w nterpolaton ponts 9 Samplng Frequence GHz - RSE/m Carrer Frequency/Hz Fg. 6. RSE of dfferent algorm n dfferent samplng frequency and nterpolaton ponts. TABLE II: RSE OF DIFFERENT ALGORITH IN DIFFERENT SAPLING FREQUENCY AND INTERPOLATION POINTS RSE/ m CF/ Hz RSE square RSE crcle RSE astersk RSE cross RSE trangle In Fg. 6, abscssa was carrer frequency and e values were Hz, Hz, 3Hz, 4Hz, 5Hz, 6Hz, 7Hz, 8Hz, 9Hz and Hz. Ordnate was RSE. - Samplng Frequence 5Hz Samplng Frequence GHz The number of nterpolaton ponts Fg. 7. RSE of dfferent algorm n dfferent numbers of nterpolaton ponts. TABLE III: RSE OF DIFFERENT ALGORITHS IN DIFFERENT NUBERS OF INTERPOLATION POINTS RSE/m Interpolaton Numbers RSE/m 9 9 Interpolaton Numbers RSE(square) RSE (crcle) In Fg. 7, abscssa was e numbers of nterpolaton ponts and e values were,9,9. Ordnate was RSE. We set e carrer frequency to 5Hz. Table III lsts e detals of each pont n e Fg. 7. From e smulaton results t could be seen at when ere were no nterpolaton ponts, e postonng accuracy was low. When we nterpolated 9 ponts to e 5Hz samplng chrps e postonng accuracy was better an e GHz samplng frequency wout nterpolaton ponts. When we nterpolated 9/9 ponts to e 74

6 5Hz/GHz samplng chrps, e postonng accuracy mproved obvously. Snce e postonng accuracy of e samplng frequency w 9 nterpolaton ponts was enough hgh, ere was lttle accuracy mprovement when we nterpolated 9 ponts to e GHz samplng chrps. 4) The postonng accuracy n dfferent samplng frequency were shown n Fg. 8: RSE/m RSE of dfferent samplng frequency - - Interpolaton pont Interpolaton pont 9 Interpolaton pont Samplng frequency/hz Fg. 8. RSE of dfferent algorm n dfferent numbers of nterpolaton ponts. TABLE IV: RSE OF DIFFERENT ALGORITHS IN DIFFERENT SAPLING FREQUENCY SF/Hz RSE/m 5 5 RSE(square) RSE (crcle) RSE (astersk) In Fg. 8, abscssa was e samplng frequency and e values were 5Hz, 5Hz, Hz (GHz). Ordnate was RSE (Unt m). We set e carrer frequency to 5Hz. Table IV lsts e detals of each pont n e Fg. 8. From e smulaton results t could be seen at e postonng accuracy was not hgh n all ree samplng frequency. When we nterpolated 9/9 ponts to e 5Hz samplng chrps e postonng accuracy was better an bo e 5Hz samplng frequency wout nterpolaton ponts and e GHz samplng frequency wout nterpolaton ponts. The accuracy of e 5Hz samplng frequency w 9/9 nterpolaton ponts was close to at of e.5ghz/5ghz samplng frequency wout nterpolaton ponts. But when e nterpolaton ponts becomes more, e mprovement of e postonng accuracy was not obvous. The postonng accuracy of 5Hz samplng frequency w 9 nterpolaton ponts was not better an e 5Hz samplng frequency w 9 nterpolaton ponts. V. CONCLUSION Ths paper ntroduced e present stuaton and future development of e small range wreless postonng, summarzed e technologes and algorms, and desgned an mproved postonng system whch proposed e mproved Snc nterpolaton algorm to reduce e samplng frequency of e system, calculated TODA values w matched flter, estmated e poston of e target node by Chan's algorm and used Chauvenet crteron to optmze e postonng results. Analyss and smulaton results showed at we could reduce e samplng frequency to a great extent wout reducng e accuracy n e mproved postonng system. The accuracy could be about cm n statc condton and blow cm n dynamc condton. So s mproved system apparently had many advantages and hgh research value. Ths proposed wreless postonng scheme also had stronger ant-nterference ablty and was easer to mplement, whch made t more promsng and was wory of contnuous research. The system ddn t need strct tme synchronzaton between e transmtter and recevers, so t ddn t need very sophstcated equpments and t was not costly compared to e exstng meods. It could be used n many felds such as warehousng, parkng lot, robot localzaton, envronment montorng, heal care, constructon, logstcs, and so on. As e demands for small range wreless postonng grew, s system would have a very broad applcaton prospects. The problem was at e postonng algorm used e mproved Snc nterpolaton algorm and we had to wat for a certan amount of chrp to form e new chrp to calculate e TDOA values. And when ere were more an one target nodes n e postonng area, ere would be nterference among e target nodes. The next work was to study multple target nodes nterference and proposed multple target nodes coordnaton algorms to confne e postonng delay and reduce and nterference. ACKNOWLEDGENT Ths work was supported by e Fundng Project for Academc Human Resources Development n Insttutons of Hgher Learnng under e Jursdcton of Bejng uncpalty under Grant PHR7, and Bejng uncpal Nature Scence Foundaton under Grant 436, and Bejng college student scence research and entrepreneural acton plan (NO. 56). REFERENCES [] T. S. Rappaport, J. H. Reed, and B. D. Wonerner, Poston locaton usng wreless communcaton on hghways of efuture, IEEE Communcatons agazne, vol. 34, pp. 33-4, October 996. [] S. Teknary, E. Chao, and R. Rchton, Performance benchmarkng for wreless locaton systems, IEEE Communcatons agazne, vol. 36, pp. 7-76, 998. [3] H. Ahn and W. Yu, Envronmental-adaptve RSSI-based ndoor localzaton, IEEE Trans. on Automaton Scence and Engneerng, vol. 6, pp , 9. [4] Y. Yu, Y. Yao, and X. Cheng, TDOA postonng technology and practcal applcaton, Chna Rado, vol., pp , 4. [5] Z. L, Z. Wang, Y. Zhang, and L. a, Precson wreless postonng scheme n small range based on frst-order dfference and correlaton nspecton, Journal of Informaton Technology Research, vol. 6, pp. -5, July-Sept. 3. [6] Y. Jang and T. Gu, Fnte leng analyss of SnC nterpolaton, Chnese Journal of Scentfc Instrument, vol. 6, pp , Sept. 5. Zhengpng L receved e doctor s degree n nformaton and communcaton engneerng at Bejng Unversty of Posts and Telecommuncatons, Bejng, Chna, n 8. He s now e assocate professor n College of Informaton Engneerng, Nor Chna Unversty of Technology, Bejng. Hs prmary role s as a tutor for e undergraduate and graduate students, cogntve 75

7 rado systems, and network economcs. He has vast experences on e followng technologes whch nclude resource management n wreless networks, meda access control, routng and QoS mechansm. Chaolang Qn s pursung master degree n nformaton and communcaton engneerng at Nor Chna Unversty of Technology, Bejng, Chna. He works on varous software development projects. Hs development skll set ncludes C/C++, Java and matlab. Hs major feld of study ncludes wreless network postonng, sgnal and nformaton processng and rado frequency dentfcaton tech. Yongme Zhang worked as a post doctor n computer nsttute at Bejng Unversty of Aeronautcs and Astronautcs, Bejng, Chna, n 8. She s now e professor n College of Informaton Engneerng, Nor Chna Unversty of Technology, Bejng. She has n-dep experences on e followng technologes whch ncludes mage processng, pattern recognton and artfcal ntellgence. She has over years of experence n software development feld. L a receved e doctor's degree n computer applcaton technology at Bejng Insttute of Technology, Bejng, Chna, n 6. He s now e professor n College of Informaton Engneerng, Nor Chna Unversty of Technology, Bejng. He has n-dep experences on e followng technologes whch ncludes advanced computng technology, grd computng, mult-agent system and embedded system. He has over years of experence n software development feld. Changlu Nu receved e doctor's degree n electromagnetc feld and electromagnetc wave currculum at Bejng Unversty of Posts and Telecommuncatons, Bejng, Chna, n 8. He s now e assocate professor n College of Informaton Engneerng, Nor Chna Unversty of Technology, Bejng. He has n-dep experences on e followng technologes whch ncludes hgh speed optcal communcaton, communcaton network and wreless sensor network. 76