Comparison and analysis of GNSS signal tracking performance based on Kalman filter and traditional loop

Transcription

1 WSEAS RANSACIONS o SIGNAL PROCESSING Xigli Su, Hoglei Qi, Jigyi Niu Compariso ad aalysis of GNSS sigal trackig performace based o Kalma filter ad traditioal loop XINGLI SUN,HONGLEI QIN,ad JINGYI NIU School of electroic ad iformatio egieerig Beihag Uiversity Xueyua Road,No.37, Haidia District,Beijig CHINA suxigli3@yahoo.com.c qhlmmm@sia.com Abstract: - As Kalma filter techology has better performace for estimatio ad predictio of dyamic sigal, it is gradually used i GNSS sigal trackig. Accordig to the steady-state error, trasfer fuctio ad equivalet oise badwidth of Kalma filter ad traditioal loop i steady status, the trackig performace of these two methods is compared i theory. he theoretical aalysis demostrates that, the dyamic stress error of Kalma filter trackig is less tha traditioal loop. Kalma filter method ca track dyamic sigal accurately with small equivalet oise badwidth. he aalysis results are verified by simulatio, ad the simulatio results show that the trackig sesitivity of Kalma filter is similar to that of the traditioal loop. he Kalma filter trackig method has higher dyamics performace ad better accuracy. Key-Words: - GNSS, Sigal rackig, Kalma Filter, traditioal loop, Phase Locked Loop, steady status. Itroductio rackig progress of GNSS receiver is essetially a problem of dyamic parameter estimatio. raditioal trackig architecture uses two loops, Delay Locked Loop (DLL) ad Phase Locked Loop (PLL). PLL is a carrier trackig loop, it adjusts the frequecy ad phase of local replicated sigal to cosist with icomig sigal. Its basic cocept is to model the icomig sigal as kow dyamics, ad determie the optimal filter structure by Wieer filter approach. he details of the traditioal PLL method ca be foud i may refereces o GNSS sigal processig []. As Kalma filter (KF) theory ca estimate ad predict dyamic sigal quite well, it is gradually used for GNSS sigal trackig process. I additio, the ewest GNSS receiver trackig algorithm, vector based trackig algorithm is realized based o KF []. herefore, usig KF to track GNSS sigal has become a mai research directio i avigatio field. here are may differet KF methods for the GNSS receiver s sigal trackig loops, differet researchers have differet ways to classify these methods, but their fudametals are the same. If classified by observatio, these methods ca be divided to two categories [3]:Kalma filter with discrimiator output as observatio; Exteded Kalma filter (EKF) with oliear observatio of I ad Q correlator outputs. Mai differece betwee them is that, the model for usig discrimiator value as observatio is liear, so it uses KF, ad that for usig basebad I ad Q correlatio values as observatio is oliear, so it uses EKF. If classified by filterig state ad observatio, they ca be divided to four categories [4]: Error state EKF for a loop filter with oliear observatio of I ad Q correlator outputs; Error state liear KF for a loop filter with discrimiator output as observatio; Error state KF for a loop filter with discrimiator output as observatio; Direct state KF for a etire sigal trackig loop with discrimiator output as observatio residuals. he mai differece betwee the secod ad third methods is that, the secod method uses discrimiator output of coheret itegratio at iitial time as observatio, ad the third method uses the average of discrimiator output durig the whole coheret itegratio process. Kalma filter used i this paper is the third method. It uses the average of discrimiator output as observatio, ad sigal parameter error as filterig state. raditioal PLL ad Kalma filter both ca realize GNSS sigal trackig, so the differece ad relatioship betwee these two methods gradually becomes a importat research poit. Certai effect has bee obtaied by usig EKF to realize digital phase locked loop (DPLL) [5]. It has bee proved that the traditioal PLL structure ca be trasformed to the similar structure of Kalma filter [6], ad E-ISSN: Issue 3, Volume 9, July 3

2 WSEAS RANSACIONS o SIGNAL PROCESSING Xigli Su, Hoglei Qi, Jigyi Niu Kalma filter also ca be trasformed to the similar structure of PLL [7]. he form of steady-state Kalma filter is similar to that of traditioal PLL. he equivalet oise badwidths ca be compared for relevat research of these two trackig methods, but it has less research [8]. I order to derive the equivalet loop badwidth of steady-state Kalma filter, steady-state Kalma gai was writte ito a form similar to the gai of PLL [9]. May mathematic approximates were made to illustrate the relatioship betwee them. However, the research results eeded to be tested ad verified. After that, the equivalet loop badwidth of EKF trackig method was obtaied by experimetal performace, ad compared with stadard PLL []. wo approaches were adopted to get it: the first was to fid the equivalet PLL that gave the same performace as the steady-state EKF, ad the secod was to fid a equivalet EKF that gave the same performace as the PLL. It was proved that Kalma filter has faster adaptio ad higher sesitivity tha stadard PLL. akig secod order carrier trackig loop for example, the carrier trackig method based o Kalma filter is researched, ad the desig method of filterig parameters is also give uder the stability coditio of steady-state Kalma filter. Accordig to the similar architecture of steady-state Kalma filter ad traditioal secod order PLL, the closed loop trasfer fuctio of steady-state Kalma filter is derived, ad its equivalet oise badwidth ad steady-state error are also calculated. By comparig above parameters with that of traditioal PLL, the reaso of Kalma filter havig better performace tha traditioal loop is aalyzed. Fially, the theoretical aalysis results are verified by simulatio. Carrier trackig model for Kalma filter Satellite sigal received by satellite avigatio receiver atea ca be expressed as follows: r( t) = dt ( ) C cos( πft + θ ) ct ( τ) + t ( ) () where dt ( ) is data code, C is carrier power, f is θ carrier frequecy icludig Doppler effects, is iitial carrier phase, ct ( ) is ragig code, τ is istataeous code phase, ad t ( ) is additive observatio oise. Ukow parameters i the model are code phase, carrier phase ad carrier Doppler frequecy. o realize satellite avigatio ad positioig, it eeds to estimate these parameters accurately, ad acquire observatio iformatio of pseudo-rage ad its rate. As satellites ad vehicle movig with time, these avigatio parameters are dyamic. he real-time estimatio process of these parameters is referred to as sigal trackig. I satellite avigatio receiver, the icomig sigal after digital samplig is correlated with a local replica to wipe-off the spreadig code, ad to accumulate sigal eergy to icrease detectability ad trackig performace. Carrier trackig is realized by correlatig with two quadrature replicas of a locally geerated carrier i two braches. After ms coheret itegral accumulatio, the models of I brach ad Q brach are expressed as follows. _ Nk Ad k m Ik ( ) = cos( θk ) R( τk + ) + Ik () _ Nk Ad k m Q ( ) = si( θ ) R( τ + ) + k k k Qk (3) where, is code offset of local referece code relative to prompt code (early code >, ad late code <), N k is the umber of samplig poits i coheret itegral time, k is the average carrier d phase amplitude i coheret itegral time, m is avigatio bit, θ k is the carrier phase error i τ coheret itegral time, k is the code phase error i coheret itegral time, Ik Qk ad are ucorrelated discrete Gaussia white oise sequeces. he phase discrimiator output of ms coheret itegral accumulatio is θ k, ad it is regarded as the average carrier phase error i itegral iterval. woquadrat arctaget phase discrimiator is used for secod order carrier phase trackig loop, ad the output of discrimiator is show as follows: QP( k) Zk ( ) = ata( ) θ ( k) + vk ( ) IP( k) (4) Where vk ( ) is radom oise of discrimiator output. he output of discrimiator Zk ( ) is the iput of Kalma filter. Basic equatios of Kalma filter are as follows: (a) project the state ahead Xˆˆk / k = Φ k, k X k (5) (b) update estimate with measuremet ˆˆˆ X k = X k / k + K k ( Z k HX k k / k ) (6) (c) compute the Kalma gai K = P H ( H P H + R ) k k / k k k k / k k k (7) (d) project the error covariace ahead _ A E-ISSN: Issue 3, Volume 9, July 3

3 WSEAS RANSACIONS o SIGNAL PROCESSING Xigli Su, Hoglei Qi, Jigyi Niu P = Φ P Φ + Q k/ k kk, k kk. k (e) update the error covariace ( ) ( ) = + k k k k / k k k k k k P I KH P I KH KRK (8) (9) Kalma filter usig i GNSS sigal trackig was realized based o these five equatios. I this paper, GNSS sigal was processed i sigal chael. How to set state X, observatio Z, observatio matrix H, oe step trasfer matrix Φ will be illustrated i this sectio. How to set observatio oise variace matrix R ad drivig oise variace matrix Q is the key poit of Kalma filter, it will be illustrated i ext sectio. Whe secod order carrier phase trackig loop of GPS sigal is realized by Kalma filter, cotiuous dyamic carrier is modeled as follows: θ θ ( ) = + wt f f () Where θ is the average carrier phase error i itegral iterval, f is the average Doppler is frequecy error i itegral iterval, ad wt ( ) Gaussia white oise vector with zero mea. Carrier phase error of discrimiator output is used as observatio of Kalma filter, the relatioship of observatio Z ad system state variable θ, f is expressed as follows: θ Z = [ ] + v f () herefore, observatio matrix is H = [ ]. Cotiuous state equatio is: x () t = F() tx() t + G () twt () () where F () t =, Gt ( ) =,ad wt () is drivig oise matrix with * dimesio. he state equatio after discretizatio is: Xk =Φ Xk + Wk (3) is Where Φ is oe step trasfer matrix, ad Wk drivig oise sequece. θk System state vector is X k =. fk Oe step trasfer matrix is: 3 3 Φ= I + F + F + F +...! 3! I + F + F! = (4) Qk = EW { k W k } is discrete drivig oise variace matrix, it ca be calculated accordig to the cotiuous drivig oise variace matrix q ( q is * matrix). 3 Qk = M+ M + M3 +...!! 3! (5) Oly adoptig the first ad secod term of (5), the approximate treatmet of discrete drivig oise variace matrix is: Qk M+ M!! ( ) ( ) = GqG + GqG F + FGqG (6) Discrete observatio of Kalma filter is Zk = θk + V k, where Vk is observatio oise sequece. 3 Parameter desig of Kalma filter trackig he Kalma filterig model of secod order carrier trackig is costat velocity (CV) model with costat coefficiet, so the carrier trackig filter is stable Kalma filter (time-ivariat or costat coefficiet Kalma filter). Comparig with ormal Kalma filter, stable Kalma filter should meet followig three coditios. (a) State model ad observatio model are time ivariat. x () t = Fx() t + Gw() t zt () = Hxt () + vt () (7) (b) Drivig oise ad observatio oise are wide stable at least. cov{ wtwt ( ) ( + τ)} = q( τ) = q δτ ( ) cov{ ( ) ( + τ)} = ( τ) = δτ ( ) vtvt R R (8) (c) Observatio iterval begis at the momet. t. Whe Kalma filter is stable, its closed loop trasfer fuctio ad equivalet oise badwidth ca be obtaied. Accordig to the drivig oise variace matrix ad observatio oise variace matrix, the state estimatio covariace matrix ad filterig gai matrix ca be calculated. Whe Kalma filter arrives to steady status, Pk = P,which is: ( )( ) P = I PH R H ΦPΦ + Q (9) he observatio of Kalma filter is carrier phase error of discrimiator output, so the observatio oise variace matrix R is the variace of carrier phase error. R = = σ δϕ C / N () E-ISSN: Issue 3, Volume 9, July 3

4 WSEAS RANSACIONS o SIGNAL PROCESSING Xigli Su, Hoglei Qi, Jigyi Niu Whe carrier oise ratio of ormal sigal is 44dB-Hz, ad the coheret itegral time is ms, the value of R is R =.4 he drivig oise variace matrix of Kalma filter maily depeds o the relative dyamic betwee satellite ad receiver ad crystal oscillator error []. Radial acceleratio of GPS satellite relative to earth surface is expressed as follows: dv d dθ a = dθ dt vr[ rr si θ ( r + r ) si θ + rr] e e s e s e s dθ = 3 dt ( r + r rr si θ e s e s ) () Where the average radius of earth is 6 re 6368km = m, the average radius of 7 satellite orbit is rs 656km =.656 m, the agular velocity of satellite is dθ π 4 =.458 rad / s, ad dt rd s θ v = dt satellite velocity is 3874 / m s he maximum radial acceleratio of GPS satellite relative to earth surface is.78 m/ s, ad its correspodig Doppler frequecy rate is: a f = f = doppl er L 8 c 3 =. 935 Hz / s = r ad / s () I GPS receiver, mai factors of crystal oscillator error ifluece o the drivig oise of Kalma filter are frequecy radom walk coefficiet h ad white oise frequecy coefficiet h []. Carrier phase oise variace caused by crystal oscillator error is: h q = ( πf ) ϕcxo (3) Doppler frequecy oise variace caused by crystal oscillator error is: q = ( πf ) π h f CXO (4) Accordig to the crystal oscillator type ad parameter of GPS receiver, q ϕ ad q f ca be calculated, ad the drivig oise variace matrix of qϕ cotiuous Kalma filter is q = ( ). qf + fdoppler Assume that the solutio of steady-state P matrix P P is P =, ad take it ito equatio (9). We P P ca get its fial solutio by method of udetermied coefficiets. Usig matrix P ca calculate the gai matrix K of Kalma filter, which is:. P P P K = PH r = = r P P r P r (5) I followig part, Kalma filter method is used to track static GPS sigal with differet power, its model is give i the first sectio, ad its observatio oise variace matrix R ad cotiuous drivig oise variace matrix q ca be calculated by ()-(4). GPS sigal is produced by simulator. he simulated sigal does ot iclude crystal oscillator error, so the desig of q igores its ifluece. With fixed values of q ad R, K ad P of Kalma filter chagig with the time are show i Fig. ad Fig.. From these figures we ca see that after Kalma filter covergece, the values of K ad P do t chage with time. Whe Kalma filter is i steady status, through equatio (9) ad (5), we ca calculate K ad P correspodig to differet observatio oise variace matrix R ad cotiuous drivig oise variace matrix q, which are show i figure 3 ad figure 4. Fig. matrix K chagig with time Fig. matrix P chagig with time E-ISSN: Issue 3, Volume 9, July 3

5 WSEAS RANSACIONS o SIGNAL PROCESSING Xigli Su, Hoglei Qi, Jigyi Niu Fig. 3 matrix K correspodig to differet q ad R Fig. 4 matrix P correspodig to differet q ad R Fig. 3 ad Fig. 4 show that uder the coditio of steady-state Kalma filter, K ad P are completely determied by R ad q. Oce R ad q are fixed, Kalma filter will coverge to the same fixed value. For traditioal PLL trackig loop, the stadard deviatio of two quadrat arctaget discrimiator output phase error caused by thermal oise is: 36 σ = (degr ee) oi se π C / N (7) For traditioal secod order PLL trackig loop, the steady-state error of two quadrat arctaget discrimiator output caused by dyamics is: d R / dt d R / dt = =. 89(degr ee) ω B σdyami c (8) Where, for GPS L bad sigal, steady-state error caused by m/ s acceleratio is: dr dt = ( m/ s ) ( 36 / cycl e) 6 ( (575. 4) cycl e/ s) / c =895 / s (9) For discrimiator output carrier phase errors of Kalma filter ad traditioal PLL, accordig to equatio (), we ca see that the steady-state errors caused by thermal oise of these two methods are same, but the steady-state errors caused by dyamic stress are differet. For differet sigal power ad vehicle dyamics, the steady-state carrier phase errors of Kalma filter ad traditioal PLL are show i figure 5. 4 heoretical trackig performace compariso ad aalysis betwee Kalma filter ad traditioal PLL 4. he steady-state error aalysis of discrimiator output Accordig to the aalysis of previous sectio, steady-state estimatio error of Kalma filter is show i equatio (9). I order to compare theoretical performace of secod order Kalma filter ad traditioal PLL by lockig loss determiatio, we compare the steady-state discrimiator output errors of these two methods. For Kalma filter trackig method, the error of carrier trackig discrimiator output is observatio residual (it is also called ew message). Uder steady-state Kalma filter status, the observatio residual is Gaussia oise with zero mea, so the variace of observatio residual error is steady-state error of Kalma filter trackig. σ = HP H + R kal ma k k / k k k (6) Fig. 5 discrimiator output carrier phase errors of Kalma filter ad traditioal PLL From Fig. 5 we ca see that, whe there is o radial dyamics betwee satellite ad receiver, discrimiator output errors of Kalma filter ad traditioal PLL have little differece, ad this error is maily determied by sigal power. Whe there is a high radial dyamics betwee satellite ad receiver, the discrimiator output error of Kalma filter is less tha that of traditioal PLL. If discrimiator output error reaches its liear rage of 9 degrees, the loop will lose lock. raditioal PLL with equivalet oise badwidth B = 5Hz loses lock whe there is 3 m/ s acceleratio dyamics, but Kalma filter does ot lose lock uder this dyamic coditio. herefore, Kalma filter has more powerful ability tha traditioal PLL for trackig dyamic sigal. E-ISSN: Issue 3, Volume 9, July 3

6 WSEAS RANSACIONS o SIGNAL PROCESSING Xigli Su, Hoglei Qi, Jigyi Niu 4. Aalysis of equivalet oise badwidth Accordig to equatio (8) we ca kow that, the steady-state error of traditioal PLL caused by dyamic stress is related to equivalet oise badwidth. herefore, we will aalyze the equivalet oise badwidth of Kalma filter, ad fid the reaso why Kalma filter has better performace tha traditioal loop. Accordig to the aalysis of the secod sectio, carrier trackig Kalma filter is stable, so we ca get its closed loop trasfer fuctio ad equivalet oise badwidth. From cotiuous Kalma filter equatio of carrier trackig loop, state estimatio value is: ˆˆˆ x() t = Fxt () + Kt ()[ zt () Hxt ()] = [ () ] ˆ() + () () F KtHxt Ktzt (9) ake Laplace trasform o both side of equatio (9), we ca get: ˆ () ( ) = + () X s si F KH KZ s (3) rasfer fuctio of stable Kalma filter is calculated as follows: ˆ H( s ) = X( s ) Z ( s ) = ( si F + KH ) K (3) Closed loop trasfer fuctio of Kalma filter is: H() s = ( si - F + KH) K Ks + K H() s s + Ks + K = = () H s K( K) s+ KK s + Ks + K (3) K = K he gai matrix of Kalma filter is K he, the closed loop trasfer fuctio of steadystate Kalma filter is: Ks + K H ( s) = s + Ks + K (33) he correspodig equivalet oise badwidth is [3]: j ds K B = H( s) H( s) = K + H ( ) j π j 4 K (34) From equatio (34) we ca kow that the equivalet oise badwidth of Kalma filter is related to the gai value of Kalma filter. For differet sigal power ad dyamics, we ca calculate differet Q ad R. By equatio (9) ad (5), we ca get the gai matrix K after Kalma filter covergece, ad the through equatio (34), we ca fially get equivalet oise badwidths of Kalma filter uder differet sigal coditios, which are show i Fig. 6. Fig. 6 Equivalet oise badwidths of Kalma filter uder differet sigal coditios From Fig. 6 we ca see that, whe sigal power becomes weaker, the equivalet oise badwidth decreases, ad whe sigal dyamics becomes higher, the badwidth icreases. herefore, the equivalet oise badwidth of Kalma filter is determied by sigal power ad dyamics. Normal GPS sigal power is always ear 44dB-Hz, ad the radial acceleratio betwee receiver ad satellite is usually less tha for 3 m/ s low dyamic vehicle. So the equivalet oise badwidth is B =. Hz for the ormal parameter of Kalma filter. Gai value ad equivalet oise badwidths of steady-state Kalma filter with typical sigal power ad dyamics are show i table. able Badwidth B ad gai K with differet Sigal power (db- sigal power ad dyamics Phase gai K Sigal dyamics (m/s Frequecy gai K Badwidth B (Hz) Hz) ) It ca be see from able that, with ormal sigal power ad dyamics, the equivalet oise badwidth of Kalma filter is less tha 5 Hz,ad it is much smaller tha the badwidth of traditioal PLL. herefore, Kalma filter trackig has better ability i oise suppressio. 4.3 Aalysis of Doppler frequecy estimatio accuracy E-ISSN: Issue 3, Volume 9, July 3

7 WSEAS RANSACIONS o SIGNAL PROCESSING Xigli Su, Hoglei Qi, Jigyi Niu Doppler frequecy estimatio error of Kalma filter ca be represeted by the frequecy dimesio value of state estimatio covariace matrix at steady status. he relatioship betwee frequecy estimatio error ad phase estimatio error of traditioal PLL is expressed as follows: κ4π BL σ = σ εf εθ 3 (35) Where the value of coefficiet κ approximates to 4 whe the equivalet oise badwidth of PLL is about Hz. he phase estimatio error of traditioal PLL is: θe σ = σ + σ + θ + PLL θ t PLL v A 3 (36) σv Where, it is take o accout to oscillator quiver caused by shake ad oscillator vibratio θ A caused by Alle deviatio. he error caused by thermal oise is: σ t PLL 36 B = π C / N (37) he error caused by dyamic stress is: d R / dt d R / dt θ = =. 89 e ω B (38) Accordig to (35)-(38), we ca calculate the theoretical Doppler frequecy accuracy of traditioal PLL. heoretical Doppler frequecy accuracies of Kalma filter ad traditioal PLL with equivalet oise badwidth 5 Hz are show i figure 7. trackig methods, i this sectio, the trackig sesitivity, dyamic adaptability ad trackig accuracy of these two methods are tested ad aalyzed by simulatio, ad the equivalet oise badwidth is used as comparig parameter. (a)rackig sesitivity aalysis he GPS data is collected from actual eviromet, ad its sigal power is reduced db for every s after 5s by addig gradually icreasig oise ito the data. Kalma filter ad traditioal PLL with differet equivalet oise badwidth are adopted to track the sigal, ad their output Doppler frequecies are show i figure 7 ad figure 8. Whe the output Doppler frequecy is wrog, the sigal power at this time is their trackig sesitivity. Fig. 8(a) rackig sesitivity of PLL Fig. 7 Doppler frequecy accuracies of KF ad PLL uder differet sigal coditios From Fig. 7 we ca see that, frequecy estimatio accuracy of Kalma filter is much higher tha that of traditioal PLL, especially i high dyamics. he mai reaso is that Kalma filter has smaller equivalet oise badwidth tha traditioal PLL uder ormal sigal coditios. 5 rackig performace simulatio compariso ad aalysis I order to validate the theoretical performace aalysis results of Kalma filter ad traditioal PLL Fig. 8(b) rackig sesitivity of Kalma filter able rackig sesitivity of Kalma filter ad PLL Badwidth B (Hz) rackig sesitivity of Kalma filter(db-hz) rackig sesitivity of PLL (db-hz) Ca t work Ca t work Ca t work It ca be see from figure 8(a), figure 8(b) ad table that, Kalma filter with ormal parameter ( B =. 7Hz ) ad traditioal PLL with ormal E-ISSN: Issue 3, Volume 9, July 3

8 WSEAS RANSACIONS o SIGNAL PROCESSING Xigli Su, Hoglei Qi, Jigyi Niu parameter ( B = 5Hz ) have approximately same trackig sesitivity 36. 5dB- H z. he Kalma filter with ormal parameter has the highest trackig sesitivity of all differet badwidths, so choosig B =. 7Hz as ormal parameter is reasoable. he situatio of PLL is ot the same. he PLL with ormal parameter B = 5Hz has lower trackig sesitivity tha that with parameter B = 3Hz. he mai reaso of choosig B = 5Hz as ormal parameter is that the trackig accuracy of B = 3Hz is obviously lower tha that of B = 5Hz (it ca be see from figure 8(a)). I additio, whe the equivalet oise badwidth is less tha 4 Hz,PLL caot work ay loger. his is because that the carrier phase error caused by crystal oscillator ad radial acceleratio betwee satellite ad receiver is too large, so the oise badwidth of PLL always larger tha 5 Hz. (b) rackig dyamic adaptability aalysis I order to test dyamic adaptability of trackig method, GPS data is produced by simulator. Its sigal power is -6dBW, ad iitial Doppler frequecy is Hz, with Hz / s ( equivalet to 3. 38m/ s )jerk all the time. he same test method is used for Kalma filter, but as Kalma filter has much better performace tha PLL i trackig dyamic sigal, so the GPS data used to test Kalma filter has a large jerk 8 Hz / s ( equivalet to m/ s ). he acceleratio of dyamic sigal gradually icreases with time. Kalma filter ad PLL with differet equivalet oise badwidths are adopted to track these sigals, ad the output Doppler frequecies are show i figure 9 ad figure. Whe the output Doppler frequecy is wrog, the acceleratio at this time is the dyamics limitatio of the trackig method. here is somethig should be iterpreted, for GPS L bad sigal, the relatioship betwee ' a f frequecy rate ad acceleratio = is, c f L Hz / s frequecy rate is equivalet to. 94m/ s acceleratio. All the simulatios below use GPS data produced by simulator, ad the acceleratio is realized by desigig frequecy rate. herefore, i the followig aalysis, frequecy rate is used to preset acceleratio dyamics. Fig. 9 dyamic adaptability of PLL trackig Fig. dyamic adaptability of Kalma filter trackig able 3 dyamic adaptability of PLL ad Kalma filter trackig Badw idth B ) ( Hz max Doppler frequecy rate of PLL trackig ( Hz / s max Doppler frequecy rate of Kalma filter trackig ( Hz / s ) ) From figure9, figure ad table3 we ca see that, Kalma filter has more powerful ability i trackig dyamic sigal tha traditioal PLL. Next, we will aalyze the reaso why Kalma filter ca track higher dyamic sigal tha traditioal PLL, accordig to the carrier phase errors of these two trackig methods with the same equivalet oise badwidth. E-ISSN: Issue 3, Volume 9, July 3

9 WSEAS RANSACIONS o SIGNAL PROCESSING Xigli Su, Hoglei Qi, Jigyi Niu Fig. discrimiator output carrier phase error of PLL rate reaches Hz / s, ad the PLL with badwidth B = Hz loses lock whe frequecy rate reaches Hz / s. he Kalma filter with badwidth B = 5Hz loses lock whe frequecy rate reaches 6833 Hz / s, ad the Kalma filter with badwidth B = Hz loses lock whe frequecy rate reaches 4938 Hz / s. herefore, with the same equivalet oise badwidth, Kalma filter has better performace i trackig dyamic sigal tha traditioal PLL. However, from figure 8 it ca be see that the frequecy accuracy of Kalma filter with large badwidth is low. Based o the aalysis of sectio 3., we kow that the Kalma filter with ormal parameter has small equivalet oise badwidth (which is.hz) uder ormal GPS sigal coditios. he badwidth is much smaller tha that of the traditioal PLL with ormal parameter (which is 5Hz). herefore, we compare the discrimiator output steady-state errors of Kalma filter ad PLL with ormal parameter i figure 3. Fig. discrimiator output carrier phase error of Kalma filter It ca be see from figure ad figure that, Kalma filter ad PLL have the same priciple of losig lock. Whe the sum of the radom error caused by thermal oise ad the steady-state error caused by dyamic exceeds 9 degrees, which is the liear rage of two-quadrat arctaget discrimiator, the loop will lose lock. Accordig to equatio (6), we ca kow that the phase error caused by thermal oise of PLL has little relatioship with loop parameters. It is maily determied by sigal power ad loop update time. he phase error of Kalma filter caused by thermal oise has the same priciple with that of PLL. However, Kalma filter i the simulatio uses fixed matrix Q ad matrix R, ad it meas the model of Kalma filter is fixed. herefore, with acceleratio gradually icreasig, the differece betwee Kalma filter model ad actual sigal is becomig larger, so the phase error caused by thermal oise is also becomig larger. Whe the acceleratio dyamics of this sigal is small, the phase error of Kalma filter caused by thermal oise is similar to that of traditioal PLL. Steady-state error caused by acceleratio dyamics of PLL is expressed as equatio (7). With acceleratio icreasig, discrimiator output steady-state error is also icreased. he PLL with badwidth B = 5Hz loses lock whe frequecy Fig. 3 Discrimiator output steady-state errors of PLL ad Kalma filter with ormal parameter From figure3 we ca see that, whe frequecy rate reaches 9 Hz / s, Kalma filter with badwidth B =. Hz loses lock. Although the equivalet oise badwidth of Kalma filter with ormal parameter is oly. Hz, it still has more powerful ability i trackig dyamic sigal tha PLL with badwidth 5 Hz. his simulatio result correspods to the aalysis results of discrimiator output steady-state error i sectio 3.. he steadystate error of Kalma filter caused by acceleratio is less tha that of traditioal PLL. his is the mai reaso why Kalma filter ca track higher dyamic sigal tha traditioal PLL. (c) rackig accuracy aalysis Kalma filter with ormal parameter ( B =. Hz ) ad traditioal PLL with B = 5Hz are adopted to track static sigal whose power is - 6dBW. he output Doppler frequecies of these two methods are show i figure4. E-ISSN: Issue 3, Volume 9, July 3

10 WSEAS RANSACIONS o SIGNAL PROCESSING Xigli Su, Hoglei Qi, Jigyi Niu Fig. 4 Doppler frequecy accuracies of PLL ad Kalma filter with ormal parameter Doppler frequecy error s stadard deviatio of Kalma filter ( B =. Hz ) is.9 Hz, ad that of traditioal PLL ( B = 5Hz ) is.33 Hz. So the Doppler frequecy accuracy of Kalma filter is much higher tha that of PLL. he mai reaso is that the less the equivalet oise badwidth of trackig method is, the higher the trackig accuracy is. 6 Coclusios I order to compare the performace of Kalma filter ad traditioal PLL trackig GPS sigals, this paper aalyses the steady-state error, closed loop trasfer fuctio ad equivalet oise badwidth of these two methods i theory. he aalysis results show that the dyamic stress error of Kalma filter is less tha traditioal PLL. Kalma filter ca track dyamic sigal accurately with lower equivalet oise badwidth. he theoretical aalysis results are also verified by simulatio. It ca be see from the simulatio results that Kalma filter has the same trackig sesitivity with traditioal PLL, ad has better dyamic adaptability ad trackig accuracy tha PLL. Kalma filter ca estimate ad predict dyamic sigal very well, so it ca take small equivalet oise badwidth to track ormal GPS sigal, ad has higher trackig accuracy. Referece [] E.D.Kapla, C.J.Hegarty, Uderstadig GPS: Priciples ad Applicatios, Artech House Publishers, secod ed. 6. [] Matthew Lashley, Modelig ad Performace Aalysis of GPS Vector rackig Algorithms, Aubur Uiversity PhD, Alabama, December 9. [3] Matthew Lashley, David M. Bevly, Joh Y.Hug, Performace Aalysis of Vector rackig Algorithms for Weak GPS Sigals i High Dyamics, Selected topics i Sigal Processig VOL.3, NO.4, August 9. [4] Jog-Hoo Wo, Domiick Dotterbock, Berd Eissfeller, Performace compariso of Differet forms of Kalma Filter Approach for a Vector-Based GNSS Sigal rackig Loop, th Iteratioal Meetig of Satellite Divisio of the istitute of Navigatio, Savaach, GA, September 9. [5] sai Sheg Kao, Sheg Chih Che, Yua Chag Chag, Exteded Kalma Filterig ad Phase Detector Modelig for a Digital Phase Locked Loop, WSEAS rasactios o Commuicatios, Issue 8, Vol. 8,9. [6] Ara Patapoutia, O phase-locked loops ad Kalma filters, IEEE rasactio o commuicatios vol.47, May 999, pp [7] Cillia O Driscoll, Mark G. Petovello, Gerard Lachapelle, Choosig the coheret itegratio time for Kalma filter-based carrier-phase trackig of GNSS sigals, GPS Solutios DOI: December. [8] Cillia O Driscoll, Gerard Lachapelle, Compariso of traditioal ad Kalma filter based trackig architectures, Europea Navigatio Coferece Naples. Italy May 9. [9] Qia Yi, Cui Xiaowei, Lu Migqua, Steadystate performace of Kalma filter for DPLL sighua Sciece ad echology Volume 4 Number 4 August 9. [] Dia Reda Salem, Cillia O Driscoll, Gérard Lachapelle, Methodology for comparig two carrier phase trackig techiques, GPS solutios, DOI:.7/s9---z,. [] Mark L. Psiaki, Hee Jug, Exteded Kalma filter methods for trackig weak GPS sigals. ION GPS, 4-7 September, Portlad, OR. []A.J.Va Dieredock, J.B.McGraw, Relatioship betwee Alla variaces ad kalma filter parameters, Proc. 6 th Aual Precise ime ad ime Iterval(PI) Applicatios ad Plaig Meetig, NASA Goddard Space Filter Ceter, 984, pp [3] Brow RG. Hwag PYG, Itroductio to radom sigals ad applied Kalma filter, 3 rd ed. Wiley, New York 997. E-ISSN: Issue 3, Volume 9, July 3