Study on GPS Common-view Observation Data with Multiscale Kalman Filter. based on correlation Structure of the Discrete Wavelet Coefficients

Transcription

1 Study on GPS Coon-view Obsevation Data with ultiscale Kalan Filte based on coelation Stuctue of the Discete Wavelet Coefficients Ou Xiaouan Zhou Wei Yu Jianguo Dept. of easueent and Instuentation, Xi dian Univesity Shaanxi, Xi an, 77,China Abstact In this pape, we pay ou attention to the upon the vaiance equation of the least squae algoith, which ultiscale alan algoith based on coelation is used to educe the ethod of estiating the ust paaete, stuctue of the discete wavelet coefficients fo the estoation of the GPS coon-view obsevation data. Based on the hypotheses of that the GPS coon-view obsevation data being peteatent possess of /f factal chaacteistics. In this condition, we estiate the ust paaete of GPS cloc diffeence data based upon the wavelet tansfo. When <<, the GPS cloc diffeence data is taen fo as a Gaussian Zeo-ean nonstationay stochastic pocess which can be consideed having the /f factal chaacteistics. So, we can tal about the coelation stuctue of the discete wavelet coefficients. Duing the couse of the estiation of the GPS coon-view data with the ultiscale alan ban, we this ust paaete is one of the paaete of GPS coon-view cloc diffeences data. << is deonstated that these discete data is a Gaussian Zeo-ean nonstationay stochastic pocess which can be consideed having the /f factal chaacteistics. In this pape, Accoding to [], the wavelet tansfo and the least squae algoith ae used to estiate ou ust paaete of the GPS data. In pape [], [3], [4], the coelation stuctue of the wavelet coefficients sequence ae taled about. Based on these chaacteistics, the ultiscale alan filte ban is designed. In section II, we expatiate the wavelet tansfo algoith of how to estiate ust paaete. In section III, on the base of ust exponent, the coelation stuctue of the pocess the single-channel and ulti-channel wavelet coefficients sequence ae discoused in evey scale. coon-view obsevation data, espectively. Copaisons between which esults and cicula T deonstate ou algoith s feasibility and effectiveness. Futheoe, in section IV, the ultiscale alan filte based on the bette ean-squae estiation is obtained. Finally, in section V, the single-channel and ulti-channel GPS coon-view obsevation data ae pocessed by the filtes, espectively. We I. INTUDUCTION give out the expeient esults. It ust be eaed that the conclusions obtained in the II. TE ESTIATION OF EXPONENT OF GPS pape is based upon the following hypotheses: CLOCK DIFFEENCE DATA ( The GPS coon-view obsevation data afte In the pape, we suppose the GPS coon-view data peteatent possess of /f factal chaacteistics. A odel fo this pocess called factional Bownian otion (fb, has been poposed by andebot and vanness. And then, the paaetes estiation of /f factal signal is tuned into the estiation of ust paaetes of fb. ( To conside the GPS coon-view cloc diffeence data pocesses is deonstated by y ( t,t I,while, the afte peteatent possess of the /f factal signal chaacte, based on this hypothesis, we can figue out its exponent. if <<, the condition of ou algoith is satisfied(that is, the GPS coon-view data afte peteatent possess of the /f factal signal chaacte, then the algoith taled about, in the pape, can be used to estiate ou cloc diffeence data. Accoding to [], we use the allat algoith to analyze discete factal Gaussian noise of the discete cloc diffeence inceent G( t y ( t y ( t data. Wavelet tansfo and the least squae algoith ae used is discete factal to estiate the exponent. Accoding to allat, In [5] Gaussian noise. This discete Gaussian noise is a Zeo-ean stationay Gaussian ando stochastic pocess. We analyze this a[] h a [], x[] g a [] ( discete factal Gaussian noise by wavelet tansfo and based /5/$. 5 IEEE. 685

2 While g and function φ (t and scale function ϕ (t g h h ae the filte coefficients of wavelet, espectively. And (. aa wavelet is the ost siply othonoal basis. Fo aa wavelet, thee is h h g g a, with this, ( tuned into [] / ( a [ ] a [ ] [] / ( a [ ] a [ ] x ( Accoding to the theoe of [6], we obtain the appoxiate wavelet coefficients a [] and detail ( (6 while -, the unnown paaetes wait fo us to estiate is (,. We get the new nonlinea equation by opeating logaith on both sides of (6 equation. ( α β log (7 ee, α log, β log. Let ( ( ( s α β log β, (8 The least squae estiation of (, β is obtained by the wavelet coefficients x [ ] by tansfoing discete factal Gaussian noise on the bases of othonoal aa discete wavelet basis accoding to allat algoith. Let [] V ae the autocoelation of the wavelet coefficients and the vaiance of the detail wavelet te,espectively. Then ( [] [], (3 least s β s, β in s( β,. So, ( α β ( log ( ( α β ( log ( ( ( V ( (4 ee, we thin the wavelet tansfoation is a whitening filte fo factal discete Gaussian noise, which have axially educed the coelation of oiginal signal. So, the coelation of wavelet coefficients obtained by wavelet tansfo can be ignoed. Because the ean value of factal discete Gaussian noise is zeo and the ean value of the wavelet coefficients afte wavelet tansfo is also zeo, the vaiance Va (x of detail wavelet te esulted fo tansfoing factal discete Gaussian on the bases of aa basis accoding to allat algoith can be estiated by the following equation: ( (5 N ( x N( While N(is the nube of detail wavelet coefficients in the th scale. Accoding to (4, we can obtain: (9 We can get α 6( α /( ( ( ( esults fo / ( We can obtain by expeient that, with the incease of the wavelet decopose scale, the estiation pecision will be elevant inceased. But the incease of the wavelet decopose scale cannot obviously ipove the estiation pecision. eanwhile, not the oe decopose scale iply the oe pecision of the estiation esults; but with the incease of the nube of the expeient points, the pecision of the estiation esults will be coesponding ipoved. III. COELATION ANALYSIS 686

3 We have detail deduced the estiation pocess of ust paaetes. Next, we will analyze the coelation of wavelet coefficients about what data we ae inteested. In [3], We now that the /f type factal pocesses ae the slow decay of the coelation stuctue associated with long -te dependencies. Fo aa wavelet, it possesses of bette coelation within scale than othe wavelets, the dependency infoation between scales and within scale will be bette used to ipove the pefoance of signal pocess. In this pape, the aa wavelet is used fo futhe study. To conside ou discete cloc diffeence data is deonstated by y (, fo [5], the wavelet seies of the pocess y ( up to the scale is given by y ( J N / / J ( a J []( n φ n J / ( N / d n n [] n ψ ( n,.n -. ( Accoding to [3] [7], the appoxiation te a J [] n of the wavelet expansion can be avoided, ( can be appoxiately denoted by ( y ψ (3 J N / / ( d [] n ( n n We conside the aa wavelet ψ ( t,, t < / / t < othewise y fo which the othonoal basis of the ultiesolution analysis is obtained. Accoding to [3], afte soe calculation, it is possible to obtain the autocoelation function of the sequence of wavelet coefficients fo n [ ] ( n E d ( n d ( With ( ( []( ξ n (4 ξ [] n [ n ] 4[ n (/ ] 6 n [ n ] 4[ n ( / ] (5 accoding to [3]and ou validation, fo (4 (5, we conclude that with the aa wavelet and /, the sequences of wavelets coefficients in each scale ae uncoelated, but when /, ( n, n >, which deonstate the wavelet coefficients is coelated. The vaiance of the wavelet coefficients is given by ( va d [] n With V ( ( V ( ( ψ ( ( (6 ψ (7 In the next section, we will use aa ultiesolution analysis and the foe equation to design ou ultiscale filtes to estiate ou cloc diffeence data. IV. BANK OF KALAN FILTES BASED ON COELATION STUCTUE OF WAVELET TE SEIES To conside the eal value of cloc diffeence at the tie of K is deonstated by χ, which constitute the state vaiable X. That is, X ( χ, The obsevation data is ( ( X ( w( y Whee, ( is obsevation atix, X( is the data of cloc diffeence to be estiated. w ( is the obsevation noise. Then the sequences of wavelet coefficients of the obsevation data is as follows : y [] n d [] n w [] n,,, J 687

4 Whee d [] n and w [] n ae the sequences of wavelet coefficients of the cloc diffeence data and obsevation noise, espectively. The sequences { [] n n N} w, ae obsevation noises with vaiance w. Accoding to [8], the sequences d [] n ae stationay pocesses fo any scale. ence, they can be appoached on the basis of the A odelin the tie-scale doain as d p [] n φ d [ n i] e [] n i i [ n ] e [ n] φ χ (8 Whee { [] n n N} e. is a zeo-ean odel noise, p denotes the ode of the A odel and the p-diensional vectos ae defined as... χ ( t [ n ] d [ n ],... d [ n p] ( φ φ φ φ,,... p (9 Accoding to Yule-Wale equation ( ( ( p ( ( ( p e a... n ( p ( p ( a by ( The optial coefficients of the alan filtes ae given φ h e h x [ n ] x [ n ] t ( h x [ n ] h (, (... ( p ( ( ( ( p ( ( ( p ( p ( p ( ee, ( n ( ( can be figue out by (4~(7. odel Based on the definition of X [] n, the state space χ [] n F χ [ n ] G [] n y e [] n [] n w [] n χ ( can be deived fo (8, whee φ φ φ p F G(,,., t (,, (3 Then, we can get the wavelet coefficients seies d [] n, by use of alan filtes, we can get [] ( χ n d [ n, n], d [ n, n ],..., d [ n, n p ] t (4 The alan algoith can be efeed by [8]. V. DATA POCESSING We pocess the single-channel and ulti-channel aw coon-view data in ou expeient, espectively. Duing the peteatent of single-channel coon-view data, the CL- NTSC in (JD(59-5 is used as expeient data. As in the pocess of peteatent, duing the tie inteval of unloced seconday planet o the inteission of coon-view, the second-degee intepolation polynoial is not used to inset absent data. Duing the peteatent of ulti-channel coon-view data, the NTSC-NICT in 5 JD( is used as expeient data, except satisfying stict coon-view equest, we delete the data of astal whose elevation less than degee. In the following, we use two steps to deonstate ou data pocessing. V.I The Estiate of Exponent As we now, the factal bown otion is a zeo-ean Gaussian nonstationay ando pocess, but its inceent is a zeo-ean stationay ando pocess, so we figue out the exponent of the factal signal by tansfoing its inceent with wavelet. ee, we ae the peteatent coon-view data as the aw factal signals, because thei ean value is not 688

5 equal to zeo, we ae zeo-ean pocess. (which cannot affect paaete. Afte soe data pocess, we can futhe obtain its inceent. That is G( t y ( t y ( t. Then, we pocess the with wavelet tansfo accoding to allat algoith. Fo the single-channel data and the ulti-channel data, we all use five scales of aa wavelet. eanwhile, we figue out the paaete use the ethod of what we have denoted. Figue (A show the inceent of the single-channel data of CL-NTSC ( JD(59-5 in. Whose paaete is Figue (B show the inceent of the ulti-channel data of NTSC-NICT(JD( in 5, whose paaete is.676. V.II The Estiation of Cloc Diffeence Data We pocess two inds of channel GPS coon-view obsevation data, espectively. Fo the white noise inseted in (4 and (6, we let it, ust as the vaiance w of the obsevation noise in obsevation equation, we also let it. The othe coefficients can be obtained by the optial coefficients deteined by (. Accoding to [3], the oiginal value of * d [] n is given by [] n y( a. That is to say, y( y( n y(3 y(4 ae ae as the oiginal value of [] n. But in ou pocessing, because of consideing coelation stuctue in ou algoith, the choice of oiginal value of d [] n does not vey ipotant fo the eventual esults. In this pape, P4. Fo single-channel data, the filte ban is coesponding to 3 scales of aa wavelet. Fo ulti-channel, 6 scales of aa wavelet ae used. Finally, the esults which only afte an inteval of five days, and at the tie point of second, the data can be eseved, ae copaed with cicula T. The final esults copaisons between CL-NTSC ( JD(59-5 and cicula T, whose oot-ean-squae is 5.ns; the esults copaisons between NTSC-NICT ( JD( and cicula T, whose oot-ean-squae is 4.89ns. The seies figues show the data befoe and afte the filte bans. VI CONCLUSION d It ust be eaed that the conclusions obtained in the pape ae based on a hypotheses. We wo out the exponent of ou coon-view data based on this hypotheses. Fo <<, we can continue to tal about the coelation stuctue of wavelet coefficients about ou data. Then we can estiate the data what we need. This algoith gives us a new ethod fo estiate the tie seies data. The othe vitue of this algoith lies in, except we let and w, othe coefficients can be obtained by theoy equation, not by expeience o tanscendent ethod, which can educe the an-ade eo by people theselves. eanwhile, which give us a good appoach to get optial coefficients value. All of these help us to get bette pecision in the final esults. EFEENCES [] Cao Kunyong, Yu Shenglin, uang xiaoqin, Paaete estiation of / ftype factal signal via wavelet tansfoation Jounal of Jilin Univesity ( Engineeing and Technology Edition vol.33.no.4.oct.3 [] A..Tewfi,. Ki Coelation Stuctue of the Discete Wavelet Coefficients of Factional Bownian otion IEEE TANS. INFO.TEOY, VOL. 38, NO., AC 99 [ 3 ] Gustavo A. ichoen, Calos E. D Attellis Estiation of Factal Signals Using Wavelets and Filte Bans IEEE TANSACTIONS ON SIGNAL POCESSING, VOL.46.NO.6.JUNE998 [4 ] P.Flandin, Wavelet analysis and synthesis of ultiesolution stochastic pocesses, IEEE TANS. INFO.TEOY,VOL,38,PP.9-97,99. [5] allat A Wavelet Tou of Signal Pocessing [6] APLAN L,KUO C J. Factal estiation fo noisy easueents via discete actional Gaussian noise(dfgn andthe aa basis[ Z]. Univ. of Southen Califonia, Los Angeles, Tech. ep. SIPI,J uly 99. [7] B.S.Chen and C.W.Lin. ultiscal Wine filte fo the estoation of factal signals:wavelet ban appoach, IEEE TANS.SIGNAL POCESSING. VOL.4,PP,97-98,994. [8] Ou Xiaouan, Zhou wei Study on GPS Coon-view Obsevation Data with ultiscale Kalan Filte Algoith 4 IEEE UFFC JOINT 5 T ANNIVESAY CONFEENCE 689

6 Figue (A CL-NTSC (JD(59-5 the inceent of single-channel cloc diffeence data Figue 3(A NTSC-NICT JD( aw ulti-channel coon-view data Figue (B NTSC-NICT(JD( the inceent of ulti-channel cloc diffeence data Figue 3(B NTSC-NICT JD( Afte the ultialan filtes consideing coelation stuctue Figue (A CL-NTSC (JD(59-5 aw single-channel coon-view data Figue (B CL-NTSC (JD(59-5 Afte the ultialan filtes consideing coelation stuctue 69