Abstract When using the signal from the Global Positioning System (GPS) for preciseestimation of a vehicle’s heading, the main challenge is finding the integer ambiguities inthe carrier phase measurements. If the distance between the receiver’s antennas, also knownas baseline length, is used in the process of solving that challenge, the resulting solution’sstability and precision can be enhanced, particularly for single-frequency L1 receivers. Thispaper presents a study of the overall influence of this baseline length in the precision andaccuracy of the heading estimation, using raw measurements from two GPS single-frequencyL1 receivers. Conclusions are presented based on the results from field trials for baselinesup to 8 m.
Vehicle navigation usually requires the determination of the course bearing and the vehicle’sheading. Due to currents and crosswinds, a vehicle’s heading is an important navigationparameter for both vessels and aircrafts. In those cases, bearing and heading angles typicallyhave to be kept different to reach a destination.
In automotive navigation this difference between course bearing and vehicle’s heading isnot significant from the navigation point of view. However, the knowledge of the heading
J. Reis · J. Sanguino (B) · A. RodriguesInstituto de Telecomunicações (IT), Instituto Superior Técnico,Technical University of Lisbon, Lisbon, Portugale-mail: [email protected]
can be used to improve the performance of road-constrained positioning algorithms, usingthe Global Positioning System (GPS), particularly near crossroads, [1]. In those cases, thealgorithms found in commercially applications traditionally delay their positioning outputuntil the vehicle clearly commits to one of the roads leaving the intersection.
Both the course bearing and a vehicle’s velocity can be estimated based on measurementsfrom a single GPS receiver, installed on the vehicle. However, inferring the vehicle’s headingfrom the velocity vector can be very difficult, particularly if the vehicle stops, as the residualerror velocities will imply that the vehicle’s heading is continuously changing.
Using GPS, the determination of the vehicle’s heading requires measurements (pseudo-ranges and/or carrier phases) from, at least, two receivers attached to it. These measurementsare used to first estimate the positioning vector from one antenna to the other (the baseline),and then its projection on a local horizontal plan would be used to compute the vehicle’sheading. Due to the GPS inherent positioning error, for short baselines, the pseudorangemeasurements alone may provide an unsatisfactory precision, [2]. However, if the signal’scarrier phase measurements are used, there is the possibility of achieving millimeter-levelprecision in the baseline vector determination, provided that the integer carrier phase ambigu-ities in these measurements are known. The process of solving these ambiguities is not trivial.For heading estimation, advantage can be taken from the knowledge of the baseline length.
In this paper, the influence of the baseline length, in the overall performance of headingestimation, is evaluated. The heading estimation was based on raw data from two single-frequency L1 GPS receivers. In the carrier phase ambiguity resolution process, the baselinelength was assumed known, and the algorithms described in [3] were used. The presentedwork was oriented towards the real-time application of the developed heading estimationalgorithms in common vehicles and, thus, only baselines up to 8 m long were studied. Fieldtrials were conducted to validate the presented results.
The rest of this paper is organized as follows. In Sect. 2, the algorithms and methods usedfor the heading estimation are presented. In Sect. 3, the hardware and testing procedures thatwere used in the field trials are described. The results are presented in Sect. 4, and finally theconclusions in Sect. 5.
2 Background Information
2.1 Baseline Vector and Heading
The attitude of a vehicle/structure is described by the Euler angles, which for the two-antennalongitudinal setup are heading (azimuth) and pitch (elevation), represented respectively byψ and θ .
Defining the baseline as the positioning vector from one antenna to the other, its localeast, north and up coordinates (e, n, u) can de related with ψ and θ by
ψ = tan−1( e
n
)(1)
and
θ = tan−1(
u√e2 + n2
)(2)
The precision of the these parameters can be easily estimated if it is assumed that there is amaximum error δ in the relative position of the baseline ends, as shown in Fig. 1.
123
Baseline Influence on Single-Frequency GPS 187
Fig. 1 Relation between relativebaseline error and attitudeangular error
The maximum angular error α is
α = tan−1(δ
L
), (3)
which, because δ is usually small when compared with L , is approximately α ≈ δL . This
means that the attitude angular error is inversely proportional to the baseline length. There-fore, for the same relative position error δ, baselines with longer L will result in a lowerangular error.
However, if δ changes with L , then the relation between the baseline length and theprecision in the heading estimate is not straight forward. In this case, δ would be a functionof L and, thus, α would no longer be inversely proportional to L .
This subject was the main motivation for the presented study and the conclusions presentedin this paper provide enlightenment regarding this point.
2.2 Single and Double Differences
Assuming that the satellite clock offset can be canceled by differentiation, the measurementof the carrier phase between receiver i and satellite k can be formulated as follows:
φki = ρk
i + cdti + λN ki + εk + εk
i , (4)
where
φki — measured carrier phase (meters),ρk
i — range between satellite k and receiver i (meters),c— speed of light in vacuum (meters per second),
N ki — integer ambiguity (cycles),εk— measurement error specific for satellite k (meters),εk
i — measurement error specific for satellite kand receiver i (meters).
With an additional measurement from a receiver (g) it is possible to create a phase SingleDifference (SD) between both receivers, by subtracting the phase measurement of receiverg, for satellite k, with the phase measurement of receiver i, for satellite k.
This can be formulated as,
φkgi = φk
g − φki =
(ρk
g − ρki
)+ c
(dtg − dti
) + λ(
N kg − N k
i
)+
(εk
g − εki
)
= ρkgi + cdtgi + λN k
gi +εkgi , (5)
where is the receiver Single Difference (SD) operator.Measurement errors specific for satellite k are the atmospheric disturbances, which are
assumed to be approximately equal because the receivers are near each other, and are thuseliminated by the SD. However, note that, in (5), the receiver clock offset is not eliminated.
123
188 J. Reis et al.
Fig. 2 GPS Interferometer fortwo observers [6]
This can be accomplished by subtracting SD’s to two different satellites, since the receiveroffsets are the same, for each SD. This Double Difference (DD or ∇�), to satellites k andm, can be written as
∇φkmgi = φk
gi −φmgi =
(ρk
gi −ρmgi
)+ λ
(N k
gi −N mgi
)+
(εk
gi −εmgi
)
= ∇ρkmgi + λ∇N km
gi + ∇εkmgi , (6)
where ∇ is the satellite Double Difference operator.The receiver clock offset is now cancelled and the phase DD is dependent on: an inte-
ger ambiguity DD, an error component and what can be called a range Double Difference.This integer ambiguity Double Difference is the true subject of the ambiguity resolution andwill thus be simply called ambiguity hereon. The above process can also be applied to thepseudorange measurements, thus creating a code Double Difference defined similarly as:
∇Pkmgi = Pk
gi −Pmgi = ∇ρkm
gi + ∇ξ kmgi , (7)
where the ∇ξ kmgi term account for the double differences on the pseudorange measurement
errors.However, due to the nature of code and phase measurements, although the code DD’s
don’t include an unknown ambiguity term, these are significantly noisier than phase DD’s.According to [4], standard deviation for code DD’s is usually about 154 times higher thanthat of phase DD’s. Besides this, the accuracy of double difference measurements also variesbetween receivers and in time.
The code and phase measurements of different satellites are assumed to be uncorrelated.However, by using Eqs. (6) and (7) to compute the covariance of the double-differences, anon-diagonal matrix is obtained, implying that double-differences are correlated [5]. Thiswill represent an important fact further ahead.
2.3 Baseline Vector and Double Differences
The relation between the baseline vector, described in Sect. 2.1, with the Single and DoubleDifferences presented in Sect. 2.2 can be seen in the next figure. Figure 2 shows a two-antennaGPS interferometer and, for each of the receivers i and g, the respective paths of propagationfor satellites k and m,as well as the range SD’s. The phase centers of the antennas are locatedat i and g. The vector b represents the baseline between the antennas phase centers, which,for this work, is to be determined, even though its length is a known fixed parameter.
123
Baseline Influence on Single-Frequency GPS 189
Due to the distance between the satellite and the two antennas, the paths of propagationare assumed to be parallel, [6]. It is clear that the projection of b onto the Line-of-Sight (LOS)between i and k equals the range SD for satellite k.
The same logic applies to satellite m, leading to:
ρkgi = b · ek, and ρm
gi = b · em, (8)
where ek and em are the LOS unit vectors for satellites k and m, commonly known as directioncosines, which, due to the distances involved, are assumed to be the same for both receivers.Thus,
∇ρkmgi = b · ekm, with ekm = ek − em. (9)
The formulation above establishes a relation between the code and phase Double Differenceswith the baseline and a direction cosine which, in its turn, depends on the known satelliteposition, as well as an estimate of the receiver’s position.
2.4 Code and Phase Observation Model
If measurements to five satellites are available, then there are four different instances of theEqs. (6) and (7), which when combined with (9) yield the following linear equation system,
with ekm, b and η being column vectors.To maximize the geometry and reduce the system’s dilution of precision, the satellite of
highest elevation angle is selected as reference, in this case satellite 1.This system of equations can be written as:
y = Bb + Aa + η, with E (y) = Bb + Aa (11)
where, E (.) is the expected value andy— vector of observed double differences, y ∈ R
m ;b— vector that contains the baseline coordinates, b ∈ R
3;a— vector of n integer double difference ambiguities, a ∈ Z
n ;B— design matrix for the baseline coordinates, B ∈ R
m×3;A— design matrix for the ambiguity terms, A ∈ R
m×n ;assuming
η ∼ N (0,Qy)— vector of unmodeled effects and measurement noise of DoubleDifferences, which is assumed to follow a Gaussian distribution withzero mean and variance Qy;
Qy— covariance matrix of the observables.
123
190 J. Reis et al.
The m × (3 + n) design matrix[
B A]
is assumed to have full rank equal to (3 + n), i.e.there are enough observations to determine the baseline coordinates and the ambiguity terms.
2.5 Introduction to the Ambiguity Resolution
As seen before, the phase DD’s include an unknown integer ambiguity term that must besolved in order compute the baseline vector, b. As there is no analytical closed form solutionfor the problem, several ambiguity resolution techniques have been proposed. The presentedmodel can be solved in a Weighted Least-Squares way that minimizes the residual error. Thisprovides a floating point solution for the ambiguities (not an integer solution).
The problem with this floating point solution is that, although it is the solution that mini-mizes the error vector given by
‖y − Aa − Bb‖2 , (12)
it does not result in the most accurate baseline coordinates since these are only available fromthe correct integer (not float) ambiguities. Therefore, it is necessary to perform a search forthese integer ambiguities.
2.6 Ambiguity Search Algorithms
The ambiguity search is the process of determining the integer vector a, of the integer doubledifference ambiguities, in (10). Typically, this process involves three steps: (i) a set is createdwith the possible (valid) candidates; (ii) a search is performed inside that set to evaluate whichof the solutions fits the best; (iii) the reached solution is validated to assure its uniquenessand correctness.
This subject has attracted numerous researchers and, currently, there are several solutionstested and proven, [7], such as: LSAST - Least-Squares Ambiguity Search Technique [8];OMEGA - Optimal Method for Estimating GPS Ambiguities [9]; FARA - Fast AmbiguityResolution Approach [10]; ARCE - Ambiguity Resolution using Constraint Equation [11]and LAMBDA - Least-Squares AMBiguity Decorrelation Adjustment [12].
2.7 The LAMBDA Method
Of all the aforementioned search methods, the LAMBDA algorithm is currently regardedas the most efficient method, also with the highest success rate [13,7]. Besides this, it alsoshowed the highest potential to be adapted to the case in which the baseline length is knownand thus the success rate and computational time are improved. Therefore, it was selected asthe solution for the search method. Although this method is extensively detailed in references[13]–[17], a short description will be provided.
Consider the observation model previously presented in Eq. (11). The matricial form ofthe this system is assumed to have full rank, thus making it possible to solve for the unknownsa and b. It is also assumed that Qy is known and positive definite. Additionally, it is importantto stress that the ambiguities are integer values and thus a ∈ Z
n whereas b ∈ R3.
With that in mind, the mathematical formulation of the least-squares problem is to find
the estimates a ∈ Z
n and
b ∈ R3 that are arguments to,
mina,b ‖ y − Aa − Bb ‖2Qy, (13)
with ‖ . ‖2Qy
= ( . )T Qy ( . ).
123
Baseline Influence on Single-Frequency GPS 191
The solution of the observation system is thus a constrained least-squares problem whichdoes not have an analytical solution. The LAMBDA search method works by performing thefollowing steps to reach an integer solution [17].
2.7.1 Ambiguity Resolution Initialization
This step consists on the definition of an initial ambiguity set. With the defined system, afloat solution represented by a ∈ R
n and b ∈ R3, can be obtained. This is accomplished
by resorting to a weighted least squares estimator, [15]. With the float solution, the integerambiguity estimator is described by the following minimization,
a = arg
(mina∈Zn
∥∥ a − a∥∥2
Qa
). (14)
This minimization cannot be solved analytically, so a search must be performed. This searchis centered on the float solution and has a range shaped by the covariance of the float solution.As mentioned before, due to the way they are obtained, the DD’s are correlated and thus thecovariance matrix of the observables is not diagonal, which in turn, means that the covari-ance matrix of the float solution is also not diagonal. Therefore, this search space won’t besymmetrical in all the dimensions.
Instead, it becomes elongated to an n-dimensional ellipsoid and thus very time-consumingto search. What distinguishes LAMBDA in this process is the search space sizing and howit is manipulated. This method uses a transformation matrix to decorrelate the error, thus di-agonalizing the float ambiguities’ covariance matrix, creating a search space that is (almost)spherical in its dimensions. This allows not only for an easier search algorithm but, mostimportantly, a greatly reduced search volume1 that translates into faster and better results.
2.7.2 Decorrelation
In order to make the float ambiguities’ covariance matrix as much diagonal as possible2, a Ztransform is applied so that the ambiguity covariance matrix can be totally decorrelated andbecome diagonal.
However, it is important to note that the elements of Z−1 are integers so that, in the finalstep of reversing the transform, original and transformed ambiguities remain integer. Thisimplies that complete diagonalization is not possible. Instead, the Z matrix is such that thenon diagonal elements of the transformed ambiguity covariance matrix are close to zero, asper the definition of “almost diagonal”.
With the decorrelation process complete, a new set of transformed float ambiguities isavailable for the search process.
This process is defined to minimize:
z = min
z
∥∥z − z∥∥2
Qz, (15)
where z = ZT a and z = ZT a.
1 Tests show that original search volumes in the order 1010–1012 can be reduced to 10–102, [15].2 In an “almost diagonal” matrix, non-diagonal elements are close to zero.
123
192 J. Reis et al.
2.7.3 Search Space Sizing
Based on the covariance matrix of the observables, an n-dimensional search space is createdaround the initial ambiguity set. With the new “almost diagonal” Euclidean norm, a searchspace of size χ is created.
This χ parameter is such that the search ellipsoid contains the required number of candi-dates, which is selected by the user. This search process is thoroughly analyzed in [16].
At the end of this search, the required number of candidates is collected and transformedback using the inverse transform, which, as mentioned earlier, transforms integers into inte-gers. The output of the algorithm is a list of candidates, sorted in ascending order of the normin (14).
2.7.4 Number of Candidates
The number of candidates must be defined by taking into account how far the floating pointsolution is from the true integer solution, as well as how much time is available for process-ing. A number of candidates too small may lead to a search volume that may not include thecorrect solution. A number too large won’t allow real-time operation.
2.8 Ambiguity Filtering
In this paper, it was chosen to process the LAMBDA method’s output in order to increaseperformance and reliability. The LAMBDA’s method success rate for single-frequency mea-surements is particularly insufficient for applications that do not allow post-processing, suchas real-time heading estimation.
In order to be able to select an ambiguity that not only is the correct one but also remainsstable, some processing was applied to the LAMBDA method’s output. After obtaining therequired number of candidates from the LAMBDA method, there’s a candidate selectionstep. In this step, the candidates are reordered using, among others, the information of theknown baseline length. The first candidate is then selected, as detailed in [3].
This candidate is further evaluated under a merit-based discrete filtering algorithm, whichoutputs a candidate with enhanced stability, [3]. With this algorithm there is no need to forcea lock on an ambiguity, and thus, with every sample, an iteration of both the LAMBDAmethod and this Ambiguity Filter is ran, requiring no post-processing to identify the correctambiguity and enhancing real-time capabilities.
This also means that every sample will be a test to the ambiguity resolution algorithm’sperformance and thus every epoch provides an evaluation of results.
3 Hardware, Equipment and Testing Procedures
In order to conduct the proposed study with real data the prototype setup represented in Fig. 3was implemented.
This setup consists of two receivers, R1 and R2, which process the electric signals receivedby two antennas, A1 and A2, which in turn are separated by a known fixed length L .
These signals are sampled and processed by the receivers, which then transmit the datato a computer. The data include raw code and carrier phase measurements, as well as, thenavigation message and the position solution, computed by the receiver.
Finally, the computer processes the data, using the methods and algorithms describedearlier, and shows real-time heading information in the screen.
123
Baseline Influence on Single-Frequency GPS 193
Fig. 3 Schematic representation of the prototype
3.1 Receivers
Two Magellan AC12 GPS receivers were used in these trials. This low-cost L1 GPS receivermeets the requirements of outputting raw code and carrier phase measurements, while pre-senting a low power consumption that allows portability.
Additionally, this receiver also features 12-channel tracking, differential GPS and SBAS,applies code-smoothing algorithms, among others, allowing it to reach a typical horizontalprecision of 1 m (CEP) and a 3 mm precision (RMS) in carrier phase measurements.
3.2 Antennas
The receivers were connected to a pair of L1 antennas: NAIS Mini Magnetic Mount Antenna.This multi-purpose small antenna provides good gain (24 dBi) while allowing a large varietyof setups in different vehicles/ structures. Its strong magnetic base permits easy and safeapplication.
For these trials, the largest baseline available of 8 m was used to measure the true heading(246,82◦) from a combination of long-term sampling and post-processing. From that base-line, a measuring tape was laid on the stone and was aligned with a laser guideline. Thissetup allowed having trials with several different baseline lengths, ranging from 1 to 8 m.The resulting baseline length measurements are accurate up to 2 mm.
Notice that, the data from the results are output from the real-time processing of thesamples, throughout 1,000 s trials.
4 Results
Table 1 shows the results for the measurements with varying baseline lengths.Regarding this table, the first column shows the baseline length. This represents the dis-
tance between the antennas’ physical centers. Besides that parameter is the number of visiblesatellites. This was included in order to validate that the test results were not being skeweddue to quite different reception conditions.
The next two columns represent, respectively, heading accuracy and precision. The firstone refers to the absolute difference in degrees between the true value of the heading and theaverage of the computed one, i.e. the accuracy. The second column consists of the 1-sigma(68% percentile) of the samples in degrees.
The average precision of these L1-only samples is 0,1◦, which is the expected precisionfor L1/L2 commercial GPS compasses. It can be seen that, the obtained precision is higher
123
194 J. Reis et al.
Table 1 Heading error (meanand variance) for varyingbaseline lengths
with a longer baseline. This was expected, according to (3), since, for a constant δ, the attitudeangular error is inversely proportional to the baseline length.
To further analyze the algorithm’s sensitivity to the baseline length it is important to eval-uate the sensitivity of the baseline vector estimation to the baseline length, i.e. to check, in(3), the dependency of δ on the baseline length.
As such, the following trials deal with the evaluation of the precision of the computedbaseline vector. Since these trials consist of different baseline lengths, it would not be pos-sible to compare the results of the baseline coordinates unless some sort of normalizationwould be applied. This would eventually result in the same effect as before, since errors inthe coordinates would be reduced or amplified with this normalization. Instead, in this paper,it is proposed to compare the length of the computed baseline vector with its true length.
Table 2 shows the baseline length computed from the algorithm’s baseline vector, for thesame trials as in Table 1.
The first column shows the 1-sigma percentile of the samples. The two other columnsshow the average baseline length measurement and the true value, as well as the respectiveabsolute difference.
It can be seen that the precision in estimate of the baseline vector does not have any definedtrend regarding the baseline length itself. Furthermore, although accuracy is also randomlyaffected by the varying baseline length, it is sized at the millimeter-level, which is quiteremarkable for a real-time single frequency interferometer. At this level of precision, smallvariations on the antenna phase centers will led to noticeable errors.
123
Baseline Influence on Single-Frequency GPS 195
5 Conclusions
In this paper were presented the results of a field trial with a real-time heading estimationalgorithm for single-frequency L1 GPS receivers.
From this work it is possible to conclude that, for baseline length values adaptable tocommon vehicles, heading precision will only be affected by these length values in the sensethat for a fixed baseline vector precision, the further the receivers are apart, the better theangular precision will be.
In fact, the algorithm’s own baseline vector precision remains unaffected when using dif-ferent baseline length values, which shows that this algorithm’s precision can be applied toall sorts of vehicles, from large road transport systems to small unmanned vehicles.
Acknowledgements This work was partially supported by the Instituto de Telecomunicaçoes (IT) and thePortuguese Foundation for the Science and Technology (FCT)—Project PTDC/EEA-TEL/122086/2010.
References
1. Simplício, P., Santos, N., Costa, A., & Sanguino, J. (2010). Uncoupled GPS road constrained posi-tioning based on constrained Kalman filtering. In Proceedings of the 13th International Symposium onWireless Personal Multimedia Communications (WPMC 2010), Recife, Brazil, 11–14 October 2010.
2. Trigo, G., Donas-Boto, D., Silva, C., & Sanguino, J. (2011). Vehicle heading estimation using twolow-cost GPS receiver configuration. In Proceedings of the 2011 IEEE 73rd Vehicular TechnologyConference (VTC 2011 Spring), Budapest, Hungary, 15–18 May 2011.
3. Reis, J. (2009). Real-time GPS heading determination, MSc thesis, Instituto Superior Técnico, Lisbon,Portugal: Technical University of Lisbon.
4. Horemuz, M., & Sjoberg, L. E. (2002). Rapid GPS ambiguity resolution for short and long base-lines. Journal of Geodesy, 76(6), 381–391.
5. Sanguino, J., Tocha, F., & Rodrigues, A. (2008). Improving positioning accuracy in WAW location-based services. In Proceedings of the IEEE International Symposium on Wireless CommunicationSystems 2008 (ISWCS’08) (pp. 123–127), Reykjavík, Iceland, October 21–24, 2008.
6. Kaplan, E. D. & Hegarty, C. J. (Eds.). (2006). Understanding GPS: Principles and applications (2nded.). Boston, MA: Artech House.
7. Kim, D., & Langley, R. B. (2000). GPS ambiguity resolution and validation: Methodologies, trendsand issues. In Proceedings of the International Symposium on GPS/GNSS, Seoul, Korea, November30 – December 2 2000.
8. Hatch, R. (1990). Instantaneous ambiguity resolution. In Proceedings of the KIS ’90, Banff, Canada,Springer-Verlag, pp. 299–308.
9. Kim, D., & Langely, R. B. (1999). An Optimized least-squares technique for improving ambiguityresolution performance and computational efficiency. In Proceedings of the ION GPS-1999, Nashville,USA.
10. Frei, E., & Beutler G. (1990). Rapid static positioning based on the fast ambiguity resolution approach“FARA”: Theory and first result. Manuscripta Geodaetica, Vol. 15, No. 4, pp. 325–356.
11. Park, C., Ilsun, K., Jang, L. G., & Jee, G. (1996). Efficient ambiguity resolution using constraintequation. In Proceedings of the IEEE PLANS, Position Location and Navigation Symposium, Atlanta,USA, pp. 277–284.
12. Teunissen, P. J. G. (1993). Least-squares estimation of the integer GPS ambiguities. Delft GeodeticComputing Centre LGR series, No. 6.
13. Buist, P. (2007). The baseline constrained LAMBDA method for single epoch, single frequency attitudedetermination applications, In Proceedings of the 20th ION International Meeting, Forth Worth, TX,pp. 2962–2973.
14. Yang, Y. (2001). Tightly integrated attitude determination methods for low-cost inertial naviga-tion: Two-antenna GPS and GPS/magnetometer, PhD Dissertation, Dept. of Electrical Engineering,University of California: Riverside, CA.
15. Borre, K., & Strang, G. (1997). Linear algebra, geodesy, and GPS. Wellesley, MA: Wellesley- Cambridge Press.
123
196 J. Reis et al.
16. Jonge, P. J., & Tiberius, C. C. J. M. (1996). The LAMBDA method for integer ambiguity estimation:Implementation aspects. Delft Geodetic Computing Centre LGR series, no. 12.
17. Jonge, P. J., Tiberius, C. C. J. M., & Teunissen, P. J. G. (1996). Computational aspects of theLAMBDA method for GPS ambiguity resolution. In Proceedings of ION GPS-96 (pp. 935–944).Missouri: Kansas City.
18. Lu, G. (1995). Development of a GPS multi-antenna system for attitude determination, PhD Thesis,University of Calgary: CA.
Author Biographies
João Reis was born in May 1985, received the degree of M.Sc. inElectrical and Computer Engineering in 2009 from Instituto SuperiorTécnico (IST), Technical University of Lisbon (UTL), having also stud-ied at the Technical University of Delft, in the Netherlands. He cur-rently works as a strategy consultant for a major telecommunicationscompany in Portugal. His research interests are navigation and satellitepositioning.
José Sanguino was born in November 1964, received the degree ofElectrical Engineer in 1989, the M.Sc. degree in Electrical and Com-puter Engineering in 1994 and the Ph.D. degree, also in Electrical andComputer Engineering, in 2004, from Instituto Superior Técnico (IST),Technical University of Lisbon (UTL). He is a researcher of the In-stituto de Telecomunicações since 1994 and he is presently an Assis-tant Professor of the Department of Electrical and Computer Engineer-ing at Instituto Superior Técnico. His current research interests involvemobile computing, wireless communications and radionavigation.
António Rodrigues received the B.Sc. and M.Sc. degrees in Electricaland Computer Engineering from the Instituto Superior Técnico (IST),Technical University of Lisbon (UTL), Lisbon, Portugal, in 1985 and1989, respectively, and the Ph.D. degree from the Catholic Univer-sity of Louvain, Louvain-la-Neuve, Belgium, in 1997. Since 1985, hehas been with the Department of Electrical and Computer Engineer-ing, IST, where is currently an Assistant Professor. His research inter-ests include mobile and satellite communications, wireless networks,spread spectrum systems, modulation and coding.
