Variable length LMS adaptive filter for carrier

Variable length LMS adaptive filter for carrier phase multipath mitigation Huicui Liu [1, 2], Xiaojing Li[2], Linlin Ge[2], Chris Rizos[2], Feixue Wang[1] 1. School of Electronic Science and Engineering, National Univ. of Defense Technology, Changsha 410073, China 2. School of Surveying and Spatial Information Systems, University of New South Wales, Sydney 2052, Australia

Abstract Multipath on carrier phase measurements are among the major error sources for short baseline positioning scenarios. To mitigate the multipath effects, in this paper adaptive Least Mean square (LMS) filters are investigated for the carrier phase data processing. The strong correlation between two consecutive days’ multipath effects on double-differenced (DD) carrier phase data is studied. Based on the fact that the antennas were static and the surrounding environment was unchangedthis fact, the multipath effects can be extracted and removed by a LMS filter, and therefore achieve higher positioning accuracy. The performances of two adaptive LMS filters, the standard LMS adaptive filter and the variable length LMS (VLLMS) adaptive filter, are analysed. Keywords: GPS, multipath, carrier phase, LMS, adaptive filter.

Introduction The multipath effect on GPS carrier phase measurements is negligible compared to the multipath error in pseudorange measurements. In Kaplan and Hegarty (2006), the maximum measurement range error due to a single multipath is stated as being less than or equal to a quarter of the carrier signal’s wavelength. This maximum is obtained when the multipath attenuation factor is less than or equal to unity, the multipath signal is orthogonal to the direct signal and the receiver delay-lock-loop maintains track on the correlation function of the direct path. However, the multipath effect is still a major error source in the case of precise static positioning. Multipath mitigation technologies or algorithms, for both pseudorange and carrier phase multipath, can be divided into three categories: antenna design, signal processing (or receiver design), and data processing. Through antenna design, such as the widely-used choke ring antenna (Fillippov, 1998; Tranquilla, 1994), multipath signals from certain directions can be blocked, and the multipath effects in both pseudorange and carrier phase are mitigated, but such antennas cannot totally remove 1

the multipath effect. Most signal processing technologies, such as MET (Townsend and Fenton, 1994) and MEDLL (Van Nee, 1994; Van Nee, 1992; Townsend, 2000), behave well for the specular pseudorange multipath signals. Compared with the first two categories, data processing techniques are a better way to address the multipath problem in carrier phase observations. Many multipath mitigation data processing techniques have been developed. For example, in Ge et al (2000) the Least Mean Square (LMS) adaptive filter was first applied for GPS multipath mitigation. However, this LMS filter had both fixed tap numbers and a fixed step-size. In this paper an enhanced LMS filter with variable length, known as a VLLMS adaptive filter (Bilcu et al, 2002), is used to mitigate multipath effects in carrier phase observations.

Multipath effects in double-differenced carrier phase observations

Satellite p Satellite q

Φ np Φ mp

Φ qm Φ qm

Antenna m Antenna n

Figure 1: Double-differenced GPS observations and its geometric relationship to a baseline

Multipath effects on carrier phase observations can be calibrated using double-differenced (DD) data. Figure 1 shows the geometric relationship of a single baseline, where m and n represent two ground antennas, while p and q represent two satellites. The carrier phase observation data of satellite i(i = p, q) received by antenna j ( j = m, n) can be written as (Kaplan and Hegarty, 2006):

Φ ij = φ ij − φ i + N ij + MPji + fτ i + fτ j − β ij + δ ij + σ ij

(1) 2

where:

φ ij is the receiver-measured signal phase, φ i is the transmitted satellite signal phase, N ij is the unknown integer number of carrier cycles, MPji is the phase error due to the multipath,

τ i and τ j are the associated satellite and receiver clock bias respectively, f is the carrier signal frequency,

β ij is the phase error due to ionospheric delay, δ ij is the phase errors due to tropospheric delay, and σ ij is the phase error due to other sources (including noise). The carrier phase DD data associated with the two satellites and the two receiver antennas can be written as: DD = (Φ mp − Φ qm ) − (Φ np − Φ qn ) = Δφ + ΔN + ΔI + ΔT + ΔMP + Δσ

(2)

With the formation of the DD, the receiver and satellite clock biases are cancelled. The remaining components are a phase term ( Δφ ) representing the linear combination of carrier phase measurements, the integer term ( ΔN ) consisting of the combined unknown integer ambiguities, multipath effect combination ( ΔMP ),a system phase noise term ( Δσ ) attributable primarily to receiver effects (Walsh, 1992), and ΔI and ΔT being the combined ionospheric and tropospheric effects respectively. All terms are expressed in units of metres. In particular, ΔMP can also be written in the form of differenced multipath effects associated with the two satellites and the two receiver antennas as in the following equation: ΔMP = ( MPmp − MPmq ) − ( MPnp − MPnq )

(3)

In order to calibrate the multipath effects on the DD carrier phase data, the baseline between the two antennas should be comparatively short, typically less than 50km (Kaplan and Hegarty, 2006), in order that the ionospheric and tropospheric effects can be assumed to largely cancel. ΔN can be assumed to be a constant if there are no cycle slips during the observation period, and it can then be removed by subtracting the mean of the DD carrier phase data. The multipath effect is the dominant error component in the DD and equation (2) can be rewritten as: 3

DD′ = DD − E [DD ] = ΔMP + ε

(4)

where ε includes Δφ and Δσ , and can be considered as noise for it is not correlated with ΔMP . The DD carrier phase data hereafter can be assumed to have mean 0. In the short baseline scenario the multipath effects on carrier phase measurements made to a certain satellite by a specific antenna can be analysed if the following conditions are satisfied: (a) One antenna (the “base antenna”) should be located in an open sky view location, and therefore assumed to be a relatively multipath-free area, while the other (the “rover antenna”) is assumed to be multipath-affected. (b) During the observation period, a satellite with a high elevation is considered to be comparatively multipath-free and serves as the “reference satellite”. (c) The two antennas are both static. With assumptions (a) and (b), and using equation (3), ΔMP is equivalent to the multipath effects of the non-reference satellite at the rover receiver antenna. With assumption (c), the multipath effect on carrier phase at some epoch is correlated with the multipath effect at the same epoch of the next sidereal day due to the fact that the azimuth and elevation of the GPS satellites and the receiver-reflectors’ locations repeat every sidereal day (approximately four minutes shorter than a mean solar day). Therefore multipath effects can be removed from carrier phase data by making use of the correlation characteristics (Ge et al, 2000). As a consequence, the positioning calculation based on the modified (multipath-mitigated) observation data can achieve higher accuracy.

The short baseline experiment A short baseline experiment was carried out to investigate the multipath effects on double-differenced carrier phase data. The experiment involved a baseline approximately 200m in length, carried out over two successive days in Centennial Park, Sydney (Australia) (Figure 2). The Base antenna was in an open (assumed comparatively) multipath-free playground area (the right photograph in Figure 2), while the Rover antenna was located near the Federal Pavilion (the left photograph in Figure 2). The experiments were carried out from about 10:00am to 12:00pm (GMT+10), on the 10th and 11th September 2008, when satellite PRN18 reached its peak elevation at this location. Two Leica GX1230 GNSS receivers were used to collect carrier phase data. The Rover antenna type was the Leica AX1202, and the Base antenna was the Leica AT504, a choke ring antenna that can mitigate the 4

multipath signals from horizontal directions. The base receiver antenna can therefore be considered as being in a relatively multipath-free area. Figure 3 shows the elevations of five satellites visible during the experiment. There were also some other satellites visible at the same time but only these five were used for positioning. Of these five satellites, PRN 18 is the only satellite during the experiment that reached its peak elevation. The DD carrier phase data associated with Rover and Base receivers are plotted in Figure 4 (in metric units), with PRN18 as the reference satellite. The first day’s data was collected from 10:24:25 to 11:22:45 (GMT+10) sampled at 0.2Hz, while the second day’s data was collected four minutes earlier. The correlation between the two days’ L1 DD carrier phase data can be clearly seen, with the correlated part mainly due to multipath effects. The overall fluctuation of PRN22’s DD data is the smallest among the four non-reference satellites because its elevation is the highest during most of the time of the experiment.

Figure 2: The short baseline experiment environment (left Rover receiver, right Base receiver).

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Figure 3: Elevation angles of the satellites during the experiment.

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Figure 4: L1 DD carrier phase data.

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The LMS adaptive filter Multipath mitigation using an LMS adaptive filter was first proposed by Ge et al (2000). The general configuration of the LMS adaptive filter is shown in Figure 5. The input signal can be represented as:

r ( n) = x ( n) + u ( n)

(5)

where x(n) is the desired signal and u (n) is the noise distortion. The three signals have the same vector length N . Assume d (n) is the reference signal.

y(n)

r (n) FIR Filter

Adaptive weight-control

e(n)

∑ + d(n)

Figure 5: LMS filter configuration.

In order to extract the relatively noise-free signal from the input signal, x(n) , u (n) and d (n) should fulfill the following conditions (Ibid, 2000): (a) The reference signal is correlated with the desired signal, such that

E[d (n) x(n − k )] = p(k ); n, k = 0,...N − 1

(6)

where p(k ) is an unknown cross-correlation for lag k. (b) The noise is correlated with neither the reference signal nor the desired signal, such that

E[ x(n)u (n − k )] = 0; n, k = 0,...N − 1

(7)

E[d (n)u (n − k )] = 0; n, k = 0,...N − 1

(8) 7

By using the LMS filter that part of the input signal which is correlated with the reference signal can be extracted. As shown in Figure 5, y (n) and e(n) are the output of the filter and the estimation error, respectively. y(n) is the part extracted from r (n) which is correlated with d (n) . In the ideal situation y(n) = x(n) . Based on the analysis in the second section, the zero-mean DD carrier phase observations in a short baseline experiment on a certain day, as shown in equation (4), can be written in the form of equation (5), where x(n) is the multipath sequence

ΔMP and u (n) is the uncorrelated noise ε . The multipath sequence for another day is then expressed as:

d (n) = x' (n) + u′(n) where x′(n) and

(9)

u′(n) have similar definitions as x(n) and u (n) . The four

vector lengths are N. Since the noise is uncorrelated with the multipath effects and the two days’ noise signals are uncorrelated with each other, x(n) , x′(n) , u (n) and u′(n) satisfy equation (6): E[d (n) x(n − k )] = E[( x′(n) + u ′(n) )x(n − k )] = E[ x′(n) x(n − k )] = p (k ); n, k = 0,...N − 1 E[ x(n)u (n − k )] = 0; n, k = 0,...N − 1 E[d (n)u (n − k )] = E[u ′(n)u (n − k )] = 0; n, k = 0,...N − 1 (10)

In this way the LMS filter can extract multipath effects (the correlated part) from DD carrier phase observations. Then in the multipath mitigation background, y(n) in Figure 5 is the extracted multipath effects and e(n) is the multipath-reduced DD carrier phase data. Also, in the ideal situation, y(n) = x(n) , which means that all the multipath effects in the input signal have been extracted.

8

In Figure 5 the filter coefficients h(n) with length M are controlled by an adaptive weight-control unit in which a least-mean-square algorithm has been introduced. In a real-time implementation h(n) can be expressed as:

hi (n + 1) = hi (n) + μe(n)r (n − i )

(11)

where μ is the step-size parameter. hi (n) ( i = 0,1..., M − 1 ) refers to the filter’s ith tap weight. n = 0,1..., N − 1 . Hence y (n) and e(n) can be expressed as: y ( n) =

M −1

∑ h ( n) r ( n − i ) i =0

i

e(n) = d (n) − y (n).

(12)

(13)

The filter length M and the step-size parameter μ should be carefully chosen to ensure that the LMS algorithm converges rapidly and is stable. Proakis (2001) and Haykin (2002) discuss the selection of the appropriate filter length and step-size parameter. In Ibid (2002), optimal filter length is discussed according to the minimum description length (MDL) criterion, which is defined as:

1 MDL( M ) = − L(θˆM ) + M ln N 2

(14)

where L(θˆM ) is the logarithm of the maximum likelihood estimates of the filter parameters. The first term tends to decrease with increasing M, while the second term increases linearly with increasing M. Hence the optimal M can balance the two trends in order to minimise MDL. In Ge et al (2000) the filter length is selected as 1 for multipath data processing. As a result of some simulations using the data collected in the experiment mentioned above, 1 is also the optimal filter length and is the value used in this paper. To ensure stability the step-size parameter μ should be in the range (Proakis, 2001): 0 m3 ⎪⎪ ⎧ m1 > m2 M 1 (k + 1) = ⎨ M 1 (k ), if ⎨ ⎩m2２ ≤ m３3 ⎪ ⎪⎩M 1 (k ) − 1, otherwise

(21)

M 2 (k + 1) = M 1 (k + 1) + 1, M 3 (k + 1) = M 1 (k + 1) + 2

(22)

z

Update the step-sizes:

μ l (k + 1) =

μN 0 M l ( K + 1)

, l = 1,2,3

(23)

where mi (i = 1,2,3) acts as the statistic square error in this algorithm. It is shown in Bilcu et al (2002) that for an uncorrelated input signal r (n) which satisfies

11

E[r (n)r (n − k )] = 0 for k ≠ 0 , the following is valid: ⎧≥ 1, if (l ) (l ) J∞ J min ⎪ = = ⎨< 1, if (m) (m) J∞ J min ⎪ 1, if ⎩

M 1 ≤ M 2 < M opt M 2 < M 1 < M opt , l = 1,2,3, m = 1,2,3, l ≠ m M 1 > M opt orM 2 > M opt

(24)

when the following relation holds at each iteration:

μ1 (k ) N1 (k ) = μ 2 (k ) N 2 (k ) = μ3 (k ) N 3 (k )

(25)

where J ∞ ＝E[e 2 (∞ ) ( l ) ], l = 1,2,3 is the steady-state MSE, J min , l = 1,2,3 is the (l )

(l )

minimum steady-state MSE, and M opt is the optimal filter length. Hence in this new algorithm, equation (21) guarantees the filter length has an optimal value. The second filter is the filter of interest in the sense that its coefficients vector will be the closest one to the Wiener filter. Selecting an optimum L is very important for the algorithm. On the one hand L should be small enough in order to update mi (i = 1,2,3) quickly. On the other hand

L should be large enough to have a sufficient number of updates. However, unfortunately there is still no theoretical optimum value for L . In this paper L is selected as 1 (based on simulation tests using the collected data). Note that equation (24) would not be valid if the input signal is correlated. However, for this application it can be assumed that the multipath sequence for carrier phase is random, i.e. uncorrelated.

Multipath mitigation on position results The L1 DD carrier phase data are shown in Figure 4, however the L2 data has similar characteristics as indicated in Figure 7. The lowest elevation satellite introduces the largest multipath effects.

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L2 DD of PRN18 and PRN6

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Figure 7: L2 DD Carrier phase DD data.

As discussed in the last section, the correlated component between successive days’ DD carrier phase data can be extracted using either the LMS filter or the VLLMS filter. There are two different filtering processes: forward filtering and backward filtering. Forward filtering means that GPS results on a given day are used to correct results on the following day, while backward filtering has the inverse meaning (Ge et al, 2000). According to Liu (2008), forward filtering is statistically better than backward filtering, and the larger the time difference between the reference and input signals the less effective the multipath mitigation will be. In this paper only forward filtering is used, and the reference signal and input signal are the DD carrier phase data of two successive days. In other words, the DD carrier phase data of the first day acts as the reference signal, and the data of the second day are the input signal. Figure 8 and Figure 9 show the DD carrier phase data for satellites PRN06 and PRN16 before and after filtering. The red line represents the input signal, which is the DD carrier phase \ data of the first day (day1) before filtering. The blue dashed and the black line represent the multipath-modified DD carrier phase data of the two filters. In these two figures, the filtering length (FL) of the LMS filter, the initial filtering length (IFL) and the statistical length L of the VLLMS filter are all unity. Only the worst multipath affected portions of the two satellite data sets are shown. From the statistical results in Table 1 and Table 2 the improvement in standard deviation of the DD carrier phase is up to 56%, and the VLLMS performs better than the LMS filter. 13

Note that both the LMS filter and the VLLMS filter have a training time, which means that the two filters can converge to the tracking state only after a finite period of time. The length of the training time, which can be validated by using a impulse sequence as the input signal and depends on the filter’s initial coefficients, is about 60 samples for this paper’s adaptive filters. So the lengths of both the input and the reference signal should be long enough to ensure that there are sufficient output samples after the filters converge.

L1 carrier phase DD of PRN06,day2/day1 0.02 0.01 0 -0.01 Before filtering After LMS filtering(FL=1) After VLLMS filtering(FL=1,L=10)

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Figure 8: DD carrier phase of PRN06 before and after filtering. Table 1: Standard deviation of DD carrier phase of PRN06 (m)

PRN06 L1 L2

LMS filtering VLLMS filtering LMS filtering VLLMS filtering

Before multipath mitigation 0.00570 0.00702

After multipath mitigation 0.00382 0.00257

Improvement 33.0% 54.9%

0.00531

24.4%

0.00350

50.1%

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L1 carrier phase DD of PRN16,day2/day1 0.015 0.01 0.005 0 -0.005 -0.01 -0.015

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Before filtering After LMS filtering(FL=1) 250 VLLMS300 350 400 After filtering(FL=1,L=10)

L2 carrier phase DD of PRN16,day2/day1 0.02 0.01 0 -0.01 -0.02 -0.03 300

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Figure 9: DD carrier phase of PRN16 before and after filtering. Table 2: Standard deviation of DD carrier phase of PRN16 (day2/day1)

PRN06 L1 L2

LMS filtering VLLMS filtering LMS filtering VLLMS filtering

Before multipath mitigation (m) 0.00508 0.00811

After multipath mitigation (m) 0.00369 0.00306 0.00476 0.00356

Improvement 27.4% 39.8% 41.3% 56.1%

The performance of the two classes of filter can also be analysed in the frequency domain. The correlation characteristics of the multipath effects on day1 and day2 can be more clearly seen from their correlation vector’s FFT. From Figure 10, where the input signal r (n) is the DD sequence from day1 and the reference signal d (n) represents the DD sequence from day2, it can be concluded that r (n) and d (n) are strongly correlated since the FFT of

Rrd

is not “white”:

Rrd (k ) = E[r (n)d (n − k )], n, k = 1,..., N .

(26)

Figure 10 also shows the FFT of Red for the two filters, where:

15

Red (k ) = E[e(n)d (n − k )], n, k = 1,2..., N .

(27)

and e(n) is the filter’s estimation error, which should be uncorrelated with d (n) . According to the experimental results shown in Figure 10, the frequency components of Rrd are mainly concentrated in the spectrum range 0~0.01Hz. However, after filtering the FFT of Red is obviously smoother, that is e(n) is more like white noise. Also, e(n) of the VLLMS filter is “whiter” than the e(n) of the LMS filter. As indicated by equation (10), the correlated components between d (n) and r (n) are the multipath effects.

PRN06 FFT of Rrd

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Figure 10: The correlation vector’s FFT between input/output signal and reference signal before and after filtering.

Because the elevation of satellite PRN18 is much higher than satellites PRN06 and PRN16, the extracted multipath effects from the two sets of DD data can be considered to be the multipath effects on the carrier phase data of satellites PRN06 and PRN16. Also, the assumption can be applied to the other two satellites: PRN15 and PRN22. Then, after cancelling the multipath effects from the carrier phase of the four lower elevation satellites, the more accurate position results based on the multipath-mitigated carrier phase data are shown in Figure 11 - Figure 13. Standard 16

deviations of the three coordinate components and the 3-D results are presented in Table 3. From these results fluctuations that existed before multipath mitigation have been suppressed, especially those in the z-coordinate component. The improvement using the VLLMS filter is greater than using the LMS filter, except for the y-coordinate component where the LMS filter performs marginally better (1.5%) than the VLLMS. About 26.3% of the multipath effects on 3-D positioning can be reduced by using the VLLMS filter. While in the case of the LMS filter this reduction is about 19.4%.

Position results of x-coordinate before multipath mitigation

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Figure 11: Position results of x-coordinate component from forward filtering (day2/day1). Table 3: Standard deviation of positioning results (day2/day1)

Standard deviation X Y Z 3-D

LMS filtering VLLMS filtering LMS filtering VLLMS filtering LMS filtering VLLMS filtering LMS filtering VLLMS filtering

Before multipath mitigation (m) 0.00359 0.00400 0.00686 0.00540

After multipath mitigation (m) 0.00334 0.00319 0.00342 0.00348 0.00495 0.00477 0.00435 0.00398

Improvement 6.96% 11.1% 14.5% 13% 27.8% 30.5% 19.4% 26.3% 17

Position results of y-coordinate before multipath mitigation 0.02

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Figure 12: Position results of y-coordinate component from forward filtering (day2/day1).

Position results of z-coordinate before multipath mitigation 0.04

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Figure 13: Position results of z-coordinate component from forward filtering (day2/day1).

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Concluding remarks The authors have introduced a multipath mitigation technique based on the VLLMS adaptive filter, and compared its performance with the LMS adaptive filter. Both filters can extract the multipath effects on carrier phase measurements. There are three steps to applying the technique. The first step is deriving the multipath-dominated DD sequences. A short baseline scenario is necessary. The rover antenna should be set up in a multipath-affected area, while the base antenna should be located in a largely multipath-free area. During the experiment a high elevation satellite is observed and serves as the reference satellite, so that the multipath effects on the DD data can be considered as the multipath effects of measurements involving the non-reference satellite and the rover antenna. The second step is filtering the DD sequences using the adaptive filter to generate the multipath effects on carrier phase. The third step is removing the multipath effects from the one-way carrier phase measurements and generating more accurate position results. Experimental results show that the performance of the VLLMS filter is better than the LMS filter’s. Up to about 26.3% of the multipath effects on the 3-D positioning can be reduced using the VLLMS filter.

References Bilcu R.C., Kuosmanen P. and Eguazarian K. (2002) A variable length LMS algorithm: Theoretical analysis and implementations, 9th International Conference on Electronics, Circuits and Systems, vol.3, 1031- 1034. Fillippov V., Tatarnicov D., Ashjaee J., Astakhov A. and Sutiagin I. (1998) The first dual-depth dual-frequency choke ring. Proceedings of the 11th International Technical Meeting of Satellite Division of the U.S. Institute of Navigation, Nashville, Tennessee, 1035-1040. Ge L., Han S.W. and Rizos C. (2000) Multipath mitigation of continuous GPS measurements using an adaptive filter. GPS Solutions, 4(2), 19-30. Haykin S.S. (2002) Adaptive Filter Theory, 4th edition, Prentice-Hall, Upper Saddle River, New Jersey. Kaplan E.D. and Hegarty C.J. (2006) Understanding GPS: Principles and Applications, 2nd edition. Artech House. Liu H., Li X., Ge L., Rizos C. and Wang F. (2008) Multipath evaluation of the SydNET CORS receivers based on pseudorange measurements. Journal of Applied Geodesy, to be published. Pritzker Z. and Feuer A. (1991) Variable length stochastic gradient algorithm. IEEE Transaction on Signal Processing, 39(4), 997-1001. Proakis J.G. (2001) Digital Communication, 4th edition, McGraw-Hill, New York, 19

Chapter 11. SydNET website, http://www.gmat.unsw.edu.au/sydnet (accessed July 2008). Townsend B. and Fenton P. (1994) A practical approach to the reduction of pseudorange multipath errors in a L1 GPS receiver. Proceedings of the 7th International Meeting of Satellite Division the U.S. Institute of Navigation, Salt Lake City, Utah, 143-148. Townsend B., Wiebe I. and Jakab A. (2000) Results and analysis of using the MEDLL receiver as a multipath meter. Proceedings of the National Technical Meeting of the U.S. Institute of Navigation, Anaheim, California, 26-29. Tranquilla J.M., Carr J.P. and Al-Rizzo H.M. (1994) Analysis of a choke ring groundplane for multipath control in global positioning system applications. IEEE Transactions on Antennas and Propagation, 42(7), 905-911. Van Nee R.D.J. (1992) The multipath estimating delay lock loop. Proceedings of the Second Internal Symposium on Spread Spectrum Techniques and Applications (ISSSTA’92), Yokohama, Japan, 39-42. Van Nee R.D.J., Siereveld J., Fenton P. and Townsend B. (1994) The multipath estimating delay lock loop: Approaching theoretical accuracy limits. Proceedings of the IEEE Position, Location and Navigation Symposium, Las Vegas, Nevada, 246-251. Walsh D. (1992) Real time ambiguity resolution while on the move. Proceedings of 5th International Technical Meeting of the Satellite Division of the U.S. Institute of Navigation, Albuquerque, New Mexico, 473-481.

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