Leaky LMS Algorithm and Fractional Brownian Motion Model for GNSS Receiver Position Estimation

Transcription

1 Leay LMS Algorithm an Fractional Brownian Motion Moel for GNSS Receier Position Estimation Jean-Philippe Montillet Enironmental Geoesy Earth Physics Research School of Earth Sciences The Australian National Uniersity Canberra, Australia Abstract This paper presents a new approach for smoothing long time series of position estimates of groun GNSS (global naigation satellite system) receiers. The fractional Brownian motion (fbm) moel is employe to escribe the position coorinate estimates that hae long-range epenencies. A new an low-complexity metho is propose to estimate the Hurst parameter an the simulation results show that the new metho achiees goo accuracy an low complexity. A moifie leay least mean squares (ML-LMS) estimator is propose to filter the long time series of the position coorinate estimates, which uses the Hurst parameter estimates to upate the filter tap weights. Simulation results emonstrate that this ML-LMS estimator outperforms the classic LMS estimator consierably in terms of both accuracy an conergence. Keywors-GNSS positioning; fractional Browinian motion moel; Hurst parameter estimation; moifie leay LMS estimator I. INTRODUCTION The Brownian motion is a well-nown moel originally use to escribe the moement of a particle in a liqui []. Due to its stochastic properties, the moel has been use in many research fiels such as in finance an engineering [, 3]. The fractional Brownian motion (fbm) base on the Hurst (H) parameter was eelope to moel long-run non-perioic statistical epenences of time series [4]. This fbm moel has been successfully applie in chemistry with the time series of the turbulences of the pressure signal of an airlift reactor [5] an in telecommunications for the stuy of specific networs [3]. In reality long time series of many processes cannot simply be moele either as Brownian motion or as Gaussian. Thus, the fbm moel gies some egree of freeom to characterize the process noise or measurement errors of the time series thans to the H parameter which was introuce in [6] for stuying the long-run non-perioic statistical epenences of time series in hyrology. This parameter is efine in the interal [0, ]. If H < 0.5 the increments of the process are negatiely correlate an the statistics of the process noise are more Gaussian, whereas if H > 0.5 the increments are positiely correlate an the statistics of the process noise are more Brownian. The fbm becomes the stanar Brownian motion when H = 0.5. Note that the fbm process exhibits long-range epenence when H > 0.5. Kegen Yu Satellite Naigation an Positioning Laboratory School of Sureying an Spatial Information Systems Uniersity of New South Wales Syney, Australia egen.yu@ieee.org This paper inestigates methos for improing the positional accuracy of groun receiers which are base on the global naigation satellite system (GNSS). The fbm moel is utilize to moel the long time series of position coorinate estimates of the GNSS receier. For simplicity, only one-imensional coorinate ata are employe in this wor for the eelopment of the theory. The egree of fitting of the fbm moel to the time series of the receier coorinate estimates will be stuie through choosing the appropriate alues of the H parameter. Knowlege about the H parameter is important an a number of ifferent methos can be use to estimate this parameter. In this paper, an alternatie technique for estimating the H parameter is propose, by winowing the ata set an using the self-similarity property of the time series. Although it is similar to the aggregate ariance metho eelope in [7], we reformulate the solution in bloc matrices. In aition, we show that the propose H-parameter algorithm reaches the imum ariance unbiase (MVU) estimator when using the Gauss-Maro moel to escribe the fluctuations of the H parameter. The MVU estimator was preiously unerline in the application of the fbm moel to networ security management [8]. The LMS algorithm has foun wie application in many areas of aaptie signal processing an control. It is a fact that the classic LMS filter is optimum when the measurement noise/error is Gaussian as explaine in [9]. Recently the leay- LMS (L-LMS) algorithms hae been propose to improe the performance of the LMS algorithm especially regaring the stability an conergence [0]. In this paper, we exploit the L- LMS algorithm to smooth the GNSS position coorinate estimates. In particular, the fbm moel an the H parameter estimation are combine with the L-LMS algorithm to form a moifie L-LMS (ML-LMS). The use of the fbm moel to aaptie filtering is relatiely new since there is only one releant reference in the literature [], where a moifie ariable step-size LMS is propose for aaptie channel equalization. Note that another approach for GNSS positional ata smoothing is the Kalman filter [], which is not consiere in this wor, but it is worth further inestigation in further wor. The remainer of the paper is organize as //$ IEEE

2 follows. The following section escribes the fbm moel an the H parameter estimation. Section III eelops the propose ML-LMS algorithm. Section IV shows simulation results base on fiel measurements, an Section V conclues the paper. II. FRACTIONAL BROWNIAN MOTION MODEL AND HURST PARAMETER ESTIMATION A. Fractional Brownian Motion Moel As mentione preiously, the fractional Brownian motion has been stuie for more than a number of ecaes. The fbm moel is efine as:, 0, / s/ /, H/, () where t enotes time, enotes the set of all the alues of a ranom function, H is the H parameter, 0, is the starting alue an, is the orinary Brownian motion. Note that t has to be positie in the general expression of the fbm. A nice oeriew of empirical stuies to calculate the H parameter can be seen in [7]. Early wors were base on the regression of the time series an the property of self-similarities was employe to compute the H parameter (e.g. [6], [3]), but this estimation metho has a high stanar eiation. By winowing the time series, [7] an [4] eelope the auto-regressie moing aerage (ARMA) an fractional auto-regressie integrate moing aerage (FARIMA) methos to compute the H. All these methos mentione aboe are generally referre to as aggregate ariance metho. More recently waelet theory has been applie to this engineering problem such as in [5]. B. Estimation of the Hurst Parameter First, let us recall the efinition of self-similarities between two functions from [4]. Definition: The increments of a ranom function { X(t, ); + > t > - } will be sai to be self-similar with parameter H (H 0) if for any h > 0 an any, {X( + h, ) - X(, )} { [X( +, ) - X(, ) ] } () This efinition relates irectly to the fbm an the increments, are stationary an self-similar with parameter H. As mentione earlier, the long-run non-perioic statistical epenences of the time series are relate to the H parameter. Thus, the approach taen in this wor to moel the single point positioning times series of the coorinates is irectly base on () an the self-similarity of the fbm. Thus, the H parameter can be extracte from (), proucing, X X log (3) X X where X(t) is the measurement ata at time t, is an arbitrary alue, h is a ariable which aries uring the ata fitting process. Clearly, H is a function of the parameters h an which are selecte by the user. In the eent that for a gien the ariable h taes ifferent alues so that it can be expresse in ector form as: h = [,,,..., ] T, < (4) Then, (3) can be rewritten in matrix form as: αh h (5) where is the column ector of all ones with length M an T H = [ H H H L ] D( α( h)) = iag{ α( h ), α( h ),, α ( hl )} α ( hi ) = (log( hi )) (6) h,) h, ) h, M ) h,) h, ) h, M ) U( h) = hl,) hl, ) hl, M ) X(+ τ ) - X() hi,) = log X(+ hiτ ) - X() where the initial time instant t is set to one for simplicity. Moreoer, the efinition of, requires that is efine in whereas h in. U then contains the time series of X with ifferent winow lengths, an (3) may be seen as an autoregressie-moing-aerage (ARMA) moel (see [9]). Thus, the H parameter can be estimate by: L Hˆ = H i (7) L i= Property : If ~α,, then gien by (7) is the imum ariance unbiase estimate of the Hurst parameter with a ariance equal to. Proof: We assume the Gauss-Maro moel of the Hurst parameter H. Then, (4) becomes H = α +W (8) where W is the error ector ue to the approximation of the Hurst parameter an it is assume to hae a Gaussian istribution with a zero mean an a coariance matrix equal to. From the parameter estimation theory [6], the performance of the imum ariance unbiase (MVU) estimator with this linear moel approaches the Cramer-Rao Lower Boun. Thus, the MVU estimate of the Hurst parameter is gien by (7) with a ariance equal to. III. MODIFIED LEAKY LMS ALGORITHM FOR SMOOTHING POSITION ESTIMATES A. Statistical moels for the aaptie filtering In this paper the time series of interest is the coorinates of a groun receier (in WGS 84) triangulate with GPS pseuoranges in single point positioning. For a fixe receier, it is interesting to moel the long-term an short-term correlation structure of these time series ata [4] when recoring the roer's position uring hours or ays of obseration. The choice of this moel is to gie some egree

3 of freeom instea of using the classic Gaussian moel which is generally use in satellite-base positioning theory [7]. One way to constrain the ariance of the receier position estimates is generally to use a Kalman filter in bacwar filtering. Here, we inten to eelop a simpler approach by using a LMS filter as the case stuy is limite to a static receier. The research community has alreay inestigate the application of the aaptie filters to constrain the error of time-of-arrial measurements in sensor localization [8]. The authors in [9] eelope a LMS filter to reuce the recurrent multipath error ue to satellite geometry on the ouble ifference obserations for a static roer. As a result, the ouble ifference resiuals hae smaller ariances an the roer's coorinate estimates are steaier. The LMS filter is an iteratie algorithm which aapts a stationary ranom process ((n)) to a esire output ((n)) with the same stochastic properties. Let us efine the iscretetime processes: = x x, = g x (9) 0 where the time series of the receier's coorinate estimates is one imensional, gien by,,...,.,, an are all in. As is the time series that we want to aapt to a esire time series, it follows that. In other wors, it can be seen that is the time series coorinate of the same roer, but filtere (e.g. moing aerage) or cog from another sensor (e.g. Inertial Measurement Unit see [0]) with smaller ariances. is the true roer's coorinate. Due to multipath propagation, an are not necessarily zero-mean Gaussian istribute, but follow a fractional Brownian motion moel as: ~ fbm ( H, σ, μ ), ~ fbm ( H, σ, μ ) (0) where an are the Hurst parameters satisfying where is a small positie number. To guarantee the conergence of the LMS filter an unbiase solution, it is important to hae a strong hypothesis on the means: where is a ery small number. In the LMS algorithm escribe in [9] the filter output is gien by where is the filter tap weight ector. The error signal is efine as so that the mean square error is gien by: T T E[ e ] = E[ ] p w + w Rw () where E[.] is the expectation operator, an T. The LMS filter is an iteratie algorithm which upates the tap weight ector accoring to: w + = w + μ e () where is a selectable parameter. It has been shown that in the case of uncorrelate Gaussian ata, conerges to the Wiener solution with 0/3 (see [0]). Each of the weight ector elements exponentially relaxes to its optimal alue with a time constant inersely proportional to the n-th eigenalue of. Further, the eigenalue sprea, efine as the ratio of the largest eigenalue oer the smallest one 0 ( ), plays a critical role in the conergence of the LMS filter. B. Moifie Leay LMS Algorithm Property : Let us consier two fbm processes with Hurst parameters an constraine by <0.5 an >0.5, an with coariance matrix an, respectiely. Then, the eigenalue spreas of the two coariance matrices satisfy: ( a) max ( a) ( b) max ( b) < (3) where the superscripts are use to enote the corresponing coariance matrices. Proof : as shown in [], the NN auto-coariance matrix of a iscrete fbm can be ecompose to be. Then, the auto-coariance matrix can be approximate as for large n where is a constant orthogonal matrix an iag{,,..., (n)}. Thus only the largest eigenalue is time epenent for large n an gien by: N ( n, H ) = Nn Γ( + ) sin H n { ( N ) N N The eigenalue sprea is then gien by max N ( n, H ) = ( H ) Nn = N + i( N i) i= N + n ( N ) N i( N i) N i= ( H ) Γ( + ) sin H (4) (5) It can be shown that the enoator in (5) basically aries between 0. an 0.48 when the H parameter ranges from 0. to 0.9. Also, the alue of the enoator when H<0.5 is greater than that when H>0.5 in general. On the other han, if N an n is large such as n, the alue of the numerator increases quicly with respect to the H parameter. Therefore, the eigenalue sprea when H<0.5 is smaller than that when H>0.5 When the eigenalue sprea increases, the rate of conergence of the LMS algorithm ecreases [0]. In our case stuy, the eigenalue sprea increases significantly as H is close to an thus it is necessary to improe the conergence rate. Similarly to the cost function efine for the ariable leay LMS algorithm in [0], our cost function is efine as J T = e + γ Hw w (6) where clearly the H parameter is irectly use to ajust the cost function. Then, the tap weight ector is upate by: w + = ( μ γ H ) w + μ e (7)

4 Clearly, the cost function in (6) is eise to achiee a goo trae-off between the optimal LMS estimator when the noise is Gaussian (H<0.5) an goo conergence when the noise is fbm (H>0.5). Note that no real ata are use through this stuy an a fbm noise function generates the receier time series. IV. SIMULATION RESULTS In this section, the noel approach to calculate the H parameter is teste against other ones preiously publishe in the scientific literature. Also, the ML-LMS algorithm eelope in Section III is ealuate through simulations using fiel measurements. A. Test of the noel metho to calculate the H parameter The fbm time series are generate using the Matlab library function wfbm (see Matlab 7.) with the true H parameter as input (first column of Table ). Then, the H parameter is estimate using our propose metho eelope in Section II- B an the results are enote by subscript p. For performance comparison, the results using two existing methos, namely the waelet metho (wa) an the auto regressie metho (RS), are also liste. Note that the estimation of the Hurst parameter with the waelet an RS methos is performe using the Matlab function wfbmesti. The results are aerage on 000 simulations an the mean an stanar eiation are extracte for each case. The results in Table clearly show that the propose metho slightly outperforms the waelet metho, whereas the performance of the RS metho is the worst. Table. Accuracy of the H parameter estimation. Statistics H B. Performances of the moifie Leay-LMSin static scenario Accoring to the moel in (0), a zero-mean Gaussian noise with stanar eiation equal to σ n = 5 m is ae to the input signal ( ) so that it has a higher stanar eiation than the reference signal ( ). The H parameters of the two time series are set to be the same an an are equal to zero. The fbm processes are generate using the Matlab library function in the same way as in Section IV-A. Thus, the ariances an are not a parameter chosen by the user. The simulation testbe uses arious H parameter alues an for each alue the error mean an stanar eiation are aerage oer 000 simulation runs. The propose leay LMS algorithm is compare against the stanar LMS metho an the original Leay LMS algorithm (L-LMS) as shown in Tables an 3. The results show that all the aaptie algorithms perform well using this testbe. The propose leay LMS (ML-LMS) an the LMS algorithm prouce similar performance for H<0.5 (in Gaussian noise), whereas if H is close to the performance of the propose algorithm is similar to that of the L-LMS. Thus the simulations confirm the mathematics eelope in the preious sections. The conergence rate is ealuate through obsering the parameter efine in the cost functions of the aaptie algorithms. It is equal to 0.(tr( Λ)) where Λ is the iagonal matrix resulting from the ecomposition of the coariance matrix of the measurements (). Fig. an Fig. show the cures of the estimation errors with the two aaptie algorithms (LMS an ML-LMS). The L-LMS is not shown here as the performances are close to the ML-LMS as seen in the preious Tables. The statistics of the aaptie filtering error is improe when using the ML-LMS. The figures confirm that the inclusion of the fbm moel in the aaptie filter helps to smooth the first an secon moment orer statistics of a time series with a ariable H parameter. Similar results were also obsere in [0]. I. CONCLUSIONS A new metho was propose to estimate the H parameter of the fbm moel that is use to escribe the long time series of the position coorinate estimates of GNSS receiers. The metho is irectly erie from the self-similarity property of the fbm. The results emonstrate that this metho significantly outperforms the RS metho in terms of the accuracy of the H parameter estimation. It also slightly outperforms the waelet metho on aerage. Also, the aaptie filter was employe for GNSS positioning an a ML- LMS algorithm was eelope to filter the long time series of the estimates of GNSS receier coorinates. The coast function of the Leay LMS is moifie with the introuction of the H parameter, resulting in the ML-LMS algorithm. The results emonstrate that the propose metho outperforms the LMS algorithm in terms of both accuracy an conergence rate. Future wor shoul be focuse on the comparison with the Kalman filter performances an on tracing moing objects. In aition, ongoing wor intens to eelop better techniques for the ML-LMS algorithm to achiee better performance. Table. Mean an stanar eiation of the three algorithms using a simulate reference series. Time Series A 0<H<0.5 Time Series B 0.5<H< Mean (m) STD (m) Mean (m) STD (m) Original Ref LMS

5 L-LMS ML-LMS Table 3. Mean an stanar eiation of the three algorithms using a meian filter. Time Series A 0<H<0.5 Time Series B 0.5<H< Mean (m) STD (m) Mean (m) STD (m) Original Me LMS L-LMS ML- LMS Fig.. Conergence time of the three aaptie algorithms when H=0.036 [3] I. Norros, On the use of fractional Brownian motion in the theory of connectionless networs, IEEE Journal on Selecte Areas of Communications, ol. 3, no. 6, pp , Aug [4] B. B. Manelbrot an J. W. Van Ness, Fractional Brownian Motions, Fractional Noises an Applications, Society for Inustrial an Applie Mathematics (SIAM) Reiew, ol. 0, no. 4, pp , Oct [5] R. Scheffer an R. Maciel Filho, The fractional Brownian motion as a moel for an inustrial airlift reactor, Chemical Engineering Science, ol. 56, no., pp , Jan. 00. [6] H. E. Hurst, R. P. Blac an Y. M. Sinaia, Long Term-Storage: An Experimental Stuy, Constable, Lonon, 965. [7] M. S. Taqqu, V. Teerosy an W. Willinger, Estimators for longrange epenence: an empirical stuy, Fractals, ol. 3, no. 4, pp , 995. Reprinte in C.J.G. Eertsz, H.-O. Peitgen, R.F. Voss (Es.), Fractal Geometry an Analysis, Worl Scientific, Singapore, 996. [8] C. M. Aujuobi, N. K. Ampah an M. N. O. Saiu, An intrusion etection technique base on change in Hurst parameter with application to networ security, International Journal of Computer Science an Networ Security, ol.7, no.5, pp , May 007. [9] S. Hayin, Aaptie Filter Theory, 4th eition, Prentice Hall Upper Sale Rier, New Jersey, 00. [0] M. Kamenetsy an B. Wirow, A Variable Leay LMS Aaptie Algorithm, Recor of the Thirty-Eighth Asilomar Conference on Signals, Systems an Computers, :5-8, Noember 004. [] A. Gupta, an S. Joshi, A noel least mean squares algorithm for tracing a iscrete-time fbm process, Proceeings of the IEEE Inian Conference, pp. -6, Sept [] W. Y. Ochieng, S. Feng, T. Moore, C. Hill an C. Hie, User leel integrity monitoring an quality control for high accuracy positioning using GPS/INS measurements, Journal of Global Positioning Systems, ol. 7, no., pp. 04-4, 008. [3] J. R. Wallis an N. C. Matalas, Small sample properties of H an K- estimators of the Hurst coefficient h, Water Resources Research, ol. 6, no. 6, pp , Dec [4] J. R. M. Hosing, Fractional ifferencing, Biometria, ol. 68, no., pp , Apr. 98. [5] S. Stoe an M. S. Taqqu, Waelet estimation for the Hurst parameter in stable processes, Lecture Notes in Physics, ol. 6, pp. 6-87, 003. [6] S. M. Kay, Funamentals of Statistical Signal Processing: Estimation Theory, PTR Prentice Hall, Englewoo Cliffs, New Jersey, 993. [7] K. Borre an G. Strang, Linear Algebra, Geoesy, an GPS, Wellesley College Publisher, 997. [8] J.-P. Montillet, K. Yu, I. Oppermann, Location performance enhancement with recursie processing of time-of-arrial measurements, Proceeings of IEEE International Symposium on Personal, Inoor an Mobile Raio Communications (PIMRC), pp. -5, Sept [9] U. Weinbach, N. Raziq an P. Collier, Mitigation of perioic GPS multipath errors using a normalise least mean square aaptie filter, Journal of Spatial Science, ol. 54, no., pp. -3, 009. [0] T. Moore, C. Hill, C. Hie, W. Ochieng, S. Feng, E. Aguao, R. Ioannies, P. Cross an L. Lau, En-to-en testing of an integrate centimetric positioning test-be, Proceeings of the ION GNSS, Fort Worth, Texas, Sept Fig.. Conergence time of the three aaptie algorithms when H=0.906 REFERENCES [] A. Papoulis an S. Unnirishna Pillai, Probability, Ranom ariables, an stochastic processes, 4th Eition, Mc Graw Hill, 00 []