Modified Leaky LMS Algorithms Applied to Satellite Positioning

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1 Modified Leaky LMS Algorithms Applied to Satellite Positioning J.P. Montillet Research School of Earth Sciences the Australian National University Australia Kegen Yu School of Geodesy and Geomatics Wuhan University China Abstract With the recent advances in the theory of fractional Brownian motion (fbm), this model is used to describe the position coordinate estimates of Global Navigation Satellite System (GNSS) receivers that have long-range dependencies. The Modified Leaky Least Mean Squares (ML-LMS) algorithms are proposed to filter the long time series of the position coordinate estimates, which uses the Hurst parameter estimates to update the filter tap weights. Simulation results using field measurements demonstrate that these proposed modified leaky least mean squares algorithms can outperform the classical LMS filter considerably in terms of accuracy (mean squared error) and convergence. We also deal with the case study where our proposed algorithms outperform the leaky LMS. The algorithms are tested on simulated and real measurements. Index Terms GNSS positioning; fractional Browinian motion model; Hurst parameter; LMS; modified leaky LMS I. INTRODUCTION The fractional Brownian motion (fbm) based on the Hurst parameter (H) was developed to model long-run non-periodic statistical dependences of time series such as in [2] and [3]. This fbm model has been successfully applied in various research topics such as telecommunications [1] or more particularly to model the noise characteristics in geodesic time series [4]. Long time series of many processes cannot simply be modelled either as Brownian motion or as Gaussian. In 1968, [3] defined the fbm model and studied the Hurst parameter. In the case of H < 0.5, the increments of the process are negatively correlated and the statistics of the process noise are more Gaussian distributed; if H > 0.5 the process exhibits longrange dependence; while the case of H = 0.5 corresponds to a pure Brownian motion (white noise). Thus, the fbm model gives some degree of freedom to characterize the process noise or measurement errors of the time series. Note that in the study of time series with coloured noise, and thus exhibit a power law frequency spectrum such as S(f) = 1/f α (see [4]), it has been shown that H is directly connected with the power-law index α such as α = 2H 1. With this definition, the random walk noise corresponds to α = 2 (H = 3/2), and white noise to α = 0 (H = 0.5). Thus, the random walk noise is classified as long-term dependency phenomena. This paper intends to improve the positional accuracy of ground receivers which are based on the global navigation satellite system (GNSS). The fbm model is utilized to model the long time series of position coordinate estimates of the GNSS receiver. For simplicity, only one-dimensional coordinate data are employed in this work for the development of the theory. The degree of fitting of the fbm model to the time series of the receiver coordinate estimates will be studied through choosing the appropriate values of the H parameter. Knowledge about the H parameter is important and a number of different methods can be used to estimate this parameter [5]. The Least Mean Squares (LMS) algorithm has found wide range of applications in many areas of adaptive signal processing and control. It is a fact that the classical LMS filter is optimum when the measurement noise/error is additive white noise as explained in [6]. Recently the Leaky LMS (L-LMS) algorithms have been proposed to improve the performance of the LMS algorithm especially regarding the stability and convergence [7]. In this paper, we exploit the L-LMS algorithm to smooth the GNSS position coordinate estimates. It follows an earlier work from [8] where a modified leaky LMS was introduced and tested via simulations. Here, the cost function of this adaptive filter is revisted to improve the performances when using real GPS time series. In Section II, we justify the use of LMS and L-LMS adaptive filters in the smoothing of estimated positions. We also develop a modified L-LMS to specifically tackle the challenges of smoothing a fbm time series with the introduction of the H parameter in the cost function of the adaptive filters. Section III is dedicated to the results when applying the theory proposed in this paper. Note that the results are based on simulated and real data obtained in static scenario. II. SMOOTHING POSITION ESTIMATES WITH LMS FILTER A. The fbm model and adaptive filtering In this paper, the time series of interest is the coordinates of a ground receiver (in World Geodetic System 1984 referential frame or also called W GS 84 [10]) either triangulated with GNSS pseudoranges in single point positioning, using double difference GNSS solutions (float ambiguities) [10], or with a new positioning technology called Locata [11]. Locata technology is a ground based positioning technologies

2 allowing point positioning of a rover with centimetre accuracy (using carrier -phase measurements) [12]. For a fixed receiver, it is interesting to model the long-term and short-term correlation structure of these time series data [9]. The choice of this model is to give some degree of freedom instead of using the Gaussian model which is generally used in satellite-based positioning theory [10]. One way to constrain the variance of the receiver position estimates is generally to use a Kalman filter in backward filtering. Here, we intend to develop a simpler approach by using a Least-Mean-Square (LMS) filter as the case study is limited to a static receiver. The research community has already investigated the application of the adaptive filters to constrain the error of time-of-arrival measurements in sensor localization [13]. The authors in [14] developed a LMS filter to reduce the recurrent multipath error due to satellite geometry on the double difference observations for a static rover. The LMS filter is an iterative algorithm which adapts a stationary random process (v(n)) to a desired output (d(n)) with the same stochastic properties. Let us define the discretetime processes: v k = x k x 0 d k = g k x 0 (1) where the time series of the receiver s coordinate estimates is one dimensional, given by x k = [ x[k], x[k + 1],..., x[k + N] ] T. xk, v k, g k and d k are all in R N. As v k is the time series that we want to adapt to a desire time series d k, it follows that σ 2 d < σ2 v. In other words, it can be seen that d k is the time series coordinate of the same rover, but filtered (e.g. moving average) or coming from another sensor (e.g. Inertial Measurement Unit see [15]) with smaller variances. x 0 is the true rover s coordinate. Due to propagation phenomena (i.e. multipath), v k and d k are not necessarily zero-mean Gaussian distributed, but follow a fbm model as: v k fbm(h v, σ 2 v, µ v ) d k fbm(h d, σ 2 d, µ d ) where H v and H d are the Hurst parameters satisfying H v H d < ɛ 1 where ɛ 1 is a small positive number. To guarantee the convergence of the LMS filter and unbiased solution, it is important to have a strong hypothesis on the means: µ v µ d < ɛ 2 where ɛ 2 is a very small number. In the LMS algorithm described in [6] the filter output is given by y k = w T k v k where w k is the filter tap weight vector. The error signal is defined as e k = d k y k so that the mean square error (MSE) is given by: (2) E{e 2 k} = E{d 2 k} 2p T w k + w T k Rw k (3) where E{.} is the expectation operator, p = E{d k v k } and R = E{v T k v k}. The LMS filter is an iterative algorithm which updates the tap weight vector w k : w k+1 = w k + 2µe k v k (4) where µ is a selectable parameter. It has been shown that in the case of data with white noise, w k converges to the Wiener solution with 0 < µ < 1/(2λ max ) with λ max the largest eigenvalue of R (see [6]). Each of the weight vector elements exponentially relaxes to its optimal value with a time constant inversely proportional to λ max. Further, the eigenvalue spread, defined as the ratio of the largest eigenvalue over the smallest one ( λmax λ min ), plays a critical role in the convergence of the LMS filter. B. Derivation of a modified Leaky-LMS to smooth position estimates It has been shown in previous works (e.g, [7]) that it is possible to improve the convergence of the LMS algorithm based on the eigenvalue spread. In other words when the eigenvalue spread increases, the rate of convergence of the LMS algorithm decreases [7]. To speed up the convergence, one way is to employ the leaky LMS filter which decreases the eigenvalue spread [6]. Regarding eigenvalue spread and a fbm process, the following property was shown in [8]. Property 2:When one considers two fbm processes with Hurst parameters H a and H b constrained by H a < 0.5 and H b > 0.5, and with covariance matrix R a and R b, respectively. Then, the eigenvalue spreads of the two covariance matrices satisfy: λ (a) max λ (a) min < λ(b) max λ (b) min where the superscripts are used to denote the corresponding covariance matrices. From [7], the cost function of the L-LMS is defined as: (5) J k = e 2 k + γw T k w k (6) where γ is the leak parameter selected by the user such as γ 0. The name stems from the fact that, when the input is turned off, the weight vector of the regular LMS algorithm stalls. With leaky LMS in the same scenario, the weight vector instead leaks out. Several works (e.g, [6], [7]) showed that: lim E{w k} = (R + γi) 1 p (7) k with this formulation, the L-LMS can be interpreted as adding zero-mean white noise with autocorrelation matrix γi to the input. The downside is that this algorithm is biased (lim k E[w k ] w op, w op optimum weight [7]). However, it can be shown that the leaky algorithm decreases the eigenvalue spread. If λ max and λ min are the largest and smallest eigenvalues of the input signal of the LMS algorithm, then the input eigenvalues seen by the leaky algorithms should be γ + λ max and γ + λ min [7]. Because of the following inequality: λ max + γ λ min + γ λ max, γ 0 (8) λ min

3 It is obvious that the eigenvalue spread of the L-LMS algorithm is smaller than that of the LMS algorithm. Similarly to the cost function defined in the variable leaky LMS algorithm, we define: J k = e 2 k + f(h)w T k w k (9) with the function f defined as: H β if H 0.5, f(h) = (H 0.5) β if > 0.5, with β 2 where clearly the H parameter is directly used to adjust the cost function. Note that if f(h) = H, the modified LMS filter is equal to the one tested in [8]. It is called in the following L LMS o. Then, the tap weight vector is updated by: w k+1 = (1 2µf(H))w k + 2µe k v k (10) Clearly, the cost function in (10) is devised to achieve a good trade-off between the optimal LMS estimator when the noise is white (H < 0.5) and good convergence when the noise is fbm (H > 0.5). This can be shown by reformulating the eigenvalue spread in Equation (8) as: λ max λ max + f(h) λ min λ min + f(h) λ max + γ λ min + γ, γ 0 (11) In other words, the inequality shows that the eigenvalue spread of the LMS algorithm may be too large, whereas the eigenvalue of the L-LMS algorithm may be too tight. On the other hand, the proposed ML-LMS algorithm with cost functions defined by (9) has moderate eigenvalue spread. In the next section, the modified L-LMS associated with the cost function in equation (9) is called ML-LMS. III. RESULTS In this section, the performance of the ML-LMS algorithm is also evaluated against the L-LMS, classical LMS and the modified L LMS o developed in [8]. A. Performances of the ML-LMS with simulated time series According to the model in (2), we generate two signals using the Matlab library function wfbm. The first one is the input signal (v k ) and the second signal is a reference signal (d k ). There are then two possible scenarios to investigate: (Scenario A) the noisy input signal is smoothed with a reference signal with a smaller noise amplitude; (Scenario B) based on integrating two different technologies at the positioning level, the two coordinate time series have different noise characteristics. Scenario A is the classical scenario of denoising a signal by adapting it with a reference signal which has similar stochastic properties. In Scenario B, the two signals are simulated with a slightly different Hurst parameter. Thus, for both scenarios µ v and µ d are small. In Scenario A, the H parameters of the two time series are set to be the same when simulating with TABLE I STATISTICS OF THE MEAN SQUARE ERROR FOR THE DIFFERENT ADAPTIVE FILTERS WITH SIMULATED TIME SERIES (SCENARIO A) MSE (m 2 ) Series 1 Series 2 0 < H < H < 1 Original Reference LMS L LMS ML LMS ML LMS o the wfbm routine. However in the input signal, we add a white noise. In Scenario B, the two signals are simulated with the wfbm routine. In order to have the standard deviation of d k smaller than the input signal for all the simulation, we arbitrarily choose σ 2 d = σ2 v/r (r in [1.3, 3]). The simulation testbed uses various H parameter values and for each value the mean square error and standard deviation are averaged over 500 simulation runs. The results in Table I are produced following the setting described in Scenario A. Overall, it shows that for simulated time series with H in [0, 0.5] or H in [0.5, 1] the LMS algorithm performs the worst in terms of mean square error. Note that for H in [0, 0.5], its performance is relatively close to the other adaptive algorithms. On the other hand, the ML- LMS achieves the minimum MSE for both Series 1 and 2. In addition, the L-LMS and ML-LMS give very similar results. The previous ML LMS o is outperformed by the M L LM S. From these simulations, one can say that a tight eigenvalue spread such as the L-LMS or the ones derived for the modified algorithm (ML-LMS) give better results than with the classical LMS. It is supported by the inequality (11) developed in the previous section. Furthermore, it is important to underline that for the values of H smaller than 0.2 and a small noise amplitude (r 1) in the input signal, then the input signal and reference are very similar. In this particular case, the LMS filter can outperform the other algorithms. This case is not included in any simulations as the values of r are chosen in [1.3, 3]. TABLE II STATISTICS OF THE MEAN SQUARE ERROR FOR THE DIFFERENT ADAPTIVE FILTERS WITH SIMULATED TIME SERIES (SCENARIO B) MSE (m 2 ) Series 1 Series 2 0 < H < H < 1 Original Reference LMS L LMS ML LMS ML LMS o Now looking at the results from the Scenario B shown in Table II, one can see that overall the statistics have values of magnitude close to 10 times greater than the results from the previous scenario. However, in this case the LMS outperforms

4 Fig. 1. Mean and standard deviation of the Error (m) between the output of the adaptive filters and the reference signal (d) following scenario A (H = 0.4, σ 2 d = σ2 v /4). Fig. 2. Mean and standard deviation of the Error (m) between the output of the adaptive filters and the reference signal (d) following scenario B (H v = 0.4, H d = 0.37, σ 2 d = σ2 v/2.3). the L-LMS, whereas the ML-LMS outperforms the LMS algorithm. This result is valid for both cases where H is in [0, 0.5] and H in [0.5, 1]. This shows when dealing with time series following the fbm model, algorithm with large eigenvalue spread can track (and smooth) the input signal more efficiently than adaptive algorithm with tighter eigenvalue spread. Another way of studying the various adaptive filters is the evaluation through observing the time series of the error between the reference signal (d) and the input signal to smooth (v). Figure 1 and 2 show the results for two different H parameter values, respectively. In both scenarios the statistics of the error is minimum for the ML-LMS. This confirms the results shown previously. In addition, one can see that the adaptive algorithms and the classical LMS differ significantly in the first 200 epochs (e.g, top graph in Figure 1 and 2). To recall the Section II-B, an eigenvalue spread too large fails to adapt the input signal with the reference one when dealing with signal experiencing a fbm model. However, a tight eigenvalue spread (i.e. L-LMS) is also not favored. In all these simulations, the ML-LMS produced the best results outperforming the LMS, L LMS and also the previous ML LMS o. This justifies the modification of the cost function of the L-LMS algorithm by introducing the H parameter. B. Application to Real data In this part, we apply the adaptive filters to two real case scenarios. The first case study is a GNSS station in the middle of London in a built-up environment. The position was recorded during 1780 epochs using Real Time Kinematic, without (v) and with a choke ring antenna (d). The position Fig. 3. East coordinates of a GNSS receiver with and withour using a choke ring antenna (top), and Mean Square Error (MSE) when using either L-LMS, ML-LMS adaptive filter. is initially triangulated in W GS 84 reference frame and then translated in a local East, North, up reference frame. The figure at the top of Figure 3 shows the time series of the East coordinate. The remaining figures display the MSE when adapting the time series of v with d using the L-LMS and ML- LMS. Note that, the results with Ml LMS o are not shown as the previous section shows that the ML-LMS outperforms the previous algorithm. The statistics of the MSE when applying the adaptive filters on the East and North coordinates, are shown in the Table III. The error is generally very small

5 TABLE III MEAN SQUARE ERROR (MSE) AND STANDARD DEVIATION FOR THE DIFFERENT ADAPTIVE FILTERS WHEN APPLIED TO THE EAST AND NORTH COORDINATE OF THE GNSS STATION FOLLOWING THE FIRST CASE STUDY. MSE (m 2 ) East North H = 0.14 H=0.22 Original Reference LMS (10 3 ) L LMS (10 3 ) ML LMS (10 3 ) TABLE IV MEAN SQUARE ERROR (MSE) AND STANDARD DEVIATION FOR THE DIFFERENT ADAPTIVE FILTERS WHEN APPLIED TO THE EAST AND NORTH COORDINATE OF THE GNSS STATION FOLLOWING THE SECOND CASE STUDY. MSE (m 2 ) East North H = 0.32 H=0.45 Original Reference LMS L LMS ML LMS as the amplitude of the time series coordinates of the GNSS receiver is in the order of magnitude of the centimetre. The overall results agree with the previous section. In particular, we can see that the statistics of the MSE corresponding to the ML-LMS are the lowest among all the adaptive filters tested here. The second case study is based on the GNSS data recorded at the station in Nottingham (UK) in open sky environment, when using pseudoranges to triangulate the position of the GNSS receiver (v). We also use the carrier-phase smoothing technique to improve the accuracy of the position of the receiver estimates (d). The statistics of the position are calculated over 1800 epochs. The statistics of the MSE when applying the adaptive filters on the East and North coordinates, are shown in the Table IV. In this case, the standard deviation of the time series of the receiver coordinates v or d is large. We can then see that the mean and standard deviation of the MSE is large compared to the first case study. Overall, the results are similar to the one before. The ML-LMS outperforms the other algorithms. Kinematic in built-up environment or using Pseudoranges. In both case studies, the ML-LMS outperform the other adaptive filters. V. ACKNOWLEDGEMENTS This research is partially supported by the Australian Research Council (grant number DP ). We also thank Dr. Lukasz K. Bonenberg from the Nottingham Geospatial Institute (NGI) at the University of Nottingham with kindly providing some data. REFERENCES [1] S. Bregni and L. Jmoda, Accurate estimation of the Hurst parameter of long-range dependent traffic using modified Allan and Hadamard variances, IEEE Transactions on Communications, vol. 56(11), , 2008 [2] H. E. Hurst, R. P. Black and Y. M. Sinaika, Long Term-Storage: An Experimental Study, Constable, London, [3] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian Motions, Fractional Noises and Applications, Society for Industrial and Applied Mathematics (SIAM) Review, vol. 10(4), , [4] J.-P. Montillet, P. Tregoning, S. McClusky, K. Yu, Extracting white noise statistics in GPS coordinate time series, IEEE Geoscience and Remote Sensing Letters, vol. 10(3), , [5] M. S. Taqqu, V. Teverovsky and W. 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CONCLUSIONS The modified Leaky LMS algorithm was developed to filter the long time series of the estimates of GNSS receiver coordinates, using a cost function as a function of the H parameter. The results demonstrated that the proposed methods outperformed the LMS algorithm in terms of MSE in the Scenario A in the simulated results. More particularly, the ML-LMS outperformed all the other algorithms in Scenario A and Scenario B.. At last, the results ended with a practical example using GNSS coordinate time series using Real time