Standard Positioning Performance Evaluation Of A Single .
(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 6, No. 3, 2015Standard Positioning Performance Evaluation of aSingle-Frequency GPS Receiver ImplementingIonospheric and Tropospheric Error CorrectionsAlban Rakipi 1, Bexhet Kamo 11Shkelzen Cakaj 1,2, Algenti Lala 12Faculty of Information TechnologyPolytechnic University of TiranaTirana, ALBANIAAbstract—This paper evaluates the positioning performanceof a single-frequency software GPS receiver using Ionosphericand Tropospheric corrections. While a dual-frequency user hasthe ability to eliminate the ionosphere error by taking a linearcombination of observables, a single-frequency user must removeor calibrate this error by other means. To remove the ionosphereerror we take advantage of the Klobuchar correction model,while for troposphere error mitigation the Hopfield correctionmodel is used. Real GPS measurements were gathered using asingle frequency receiver and post–processed by our proposedadaptive positioning algorithm. The integrated Klobuchar andHopfield error correction models yeild a considerable reductionof the vertical error. The positioning algorithm automaticallycombines all available GPS pseudorange measurements whenmore than four satellites are in use. Experimental results showthat improved standard positioning is achieved after errormitigation.Keywords—algorithm; Global Positioning System; GDOP;Hopfield model; Klobuchar model; receiver; PVT; RawmeasurementsI.INTRODUCTIONRecently, there is an increase interest in positioningtechniques based on GNSS (Global Navigation SatelliteSystems) such as GPS (Global Positioning System). GPS is asatellite-based navigation radio system which is used to verifythe position and time in space and on the Earth [1]. Thestandard approach for estimating the receiver position andclock offset is first to linearize the pseudorange measurementsaround a rough guess of the receiver position and clock biasand then to iterate until the difference between the guess andthe measurements approaches zero. While this implies thatsome information is needed about the initial receiver position,it turns out that the solution is not very sensitive to this initial(or rough) guess [2]. The GPS satellites are orbiting the Earthat altitudes of about 20.200 km and it is generally known thatthe atmospheric effects on the GPS signals are the mostdominant spatially correlated biases. The atmosphere causingthe delay in GPS signals consists of two main layers:ionosphere and troposphere [3].The Ionosphere is the band of the atmosphere from around(50 – 1000 km) above the earth’s surface and is highlyvariable in space and time, with certain solar-relatedionospheric disturbances [4]. Ionosphere research attractsFaculty of Electrical and Computing EngineeringPrishtina UniversityPrishtina, KOSOVOsignificant attention from the GPS community becauseionosphere range delay on GPS signals is a major error sourcein GPS positioning and navigation. The ionospheric delay is afunction of the total electron content (TEC) along the signalpath and the frequency of the propagated signal, mostlyaffecting the vertical component of user’s position. Two mainstatistical model are available for the correction of ionosphericrange error in single frequency applications: the Klobucharmodel for GPS [3] or the NeQuick model [2] foreseen for usein European GALILEO system.The troposphere is the band of the atmosphere from theearth’s surface to about 8 km over the poles and 16 km overthe equator [5]. The tropospheric propagation delay is directlyrelated to the refractive index (or refractivity). The signalrefraction in the troposphere is separated into twocomponents: the dry and the wet component, where the dry orhydrostatic component is mainly a function of atmosphericpressure and gives rise to about 90% of the tropospheric delay.There are different mathematical models that can be used tocorrect the tropospheric error such as Saastamoinen andHopfield Model [6].The paper is organized into seven major sections. The firstsection goes over background on positioning techniques basedon GNSS and atmospheric errors. The second sectiondescribes the data collection process and the tools used formeasurements. The third section is referred to acquisition andtracking of GPS signals. The fourth section gives a high leveldescription of our approach in implementing the positioningalgorithm. Section five is dedicated to Ionospheric andTropospheric error correction models focusing on Klobucharand Hopfield models. Section six presents some resultsobtained analyzing the algorithm performance with andwithout error corrections. Finally the last section draws theconclusions.II.DATA COLLECTION PROCESSIn this section is described the GPS data collection processand the implementation of a post-processing adaptive PositionVelocity Time (PVT) algorithm, where we includedmathematical Ionospheric and Tropospheric correction modelsaiming to an improved accuracy of user’s position estimation.An experiment was conducted using GPS C/A-codepseudorange data collected outside our laboratory in the27 P a g ewww.ijacsa.thesai.org
(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 6, No. 3, 2015Polytechnic University of Tirana Campus. The preciseCartesian coordinates of our stationary receiver werepreviously determined by a professional receiver for latercomparison. Totally 2340 epochs of data were analyzed forthe experiment and post-processed in Matlab environment. Inthe following subsections we describe the tools used formeasurements.A. Receiver UnitIn this section we give a brief description of the receiverused to collect the data and of the software used to processthem. SAT-SURF [7] is a hardware black-box integrating GPSand GSM/GPRS functionalities. It appears as a metallic boxwith two external antennas, a USB cable and a power supplycable (Fig. 1). The SAT-SURF hardware allows getting outfrom the GPS receiver several data and also each availableraw measurement (depending on the receiver capabilities).Each GPS parameter is logged with a related GPS time stamp,so that each parameter can be aligned to the evolution of allthe others. In our measurements we used SAT-SURF with thecore components indicated in Table 1.SAT-SURFER [7] is the software suite running on astandard PC that uses data coming from SAT-SURF. It is asoftware suite able to talk in real-time with state-of-the-artGPS receiver modules as well as external professional GPSunits. SAT-SURFER gets raw data, displays such data on thescreen and log them in different files allowing any postprocessing activity.Fig. 1. SAT-SURF: view of the case (a) and of the hardware board (b)TABLE I.ParameterSensitivityTTFFAugmentationUpdate rateOtherSAT-SURF SUB-SYSTEM TECHNICAL SPECIFICATIONSValue-160dBmUnder 1 second Time-To-First-Fix for Hot and AidedStartsSupports SBAS: WAAS, EGNOS, MSAS, AssistedGPS4 HzGalileo-ready receiverB. Error correction parametersThe major error contribution in the overall user positionaccuracy comes from the Ionosphere layer, affecting mostlythe vertical component and increasing in such way VDOP(Vertical Dilution of Precision) [8]. The ionosphericparameters taken from the SAT-SURFER log files areilustrated in Fig.2.Fig. 2. Ionospheric correction parameters taken from SAT-SURFER log fileThe α and β are the input data of our adaptive positioningalgorithm necessary for the mitigation of ionospheric error inthe user’s position estimation. It will be later shown that weachieve a considerable improvement of the vertical componentand a decreased VDOP, after the application of this correctionin the main algorithm.III.GPS SIGNAL ACQUISITION AND TRACKINGThe purpose of acquisition is to determine coarse values ofcarrier frequency and code phase of the satellite signals. Manyresearch works focus on base-band signal processing in thesoftware receivers. [9].There are several acquisition methods for GPS signalsintroduced in recent years, which are often implemented intime domain and frequency domain. Among these methods,serial search acquisition is a traditional method for acquisitionin CDMA system, but it is time-consuming and performedthrough hardware in the time domain. In contrast, theconventional parallel in frequency method increases the speedof acquisition by transforming correlation calculation into thefrequency domain through DFT (Discrete Fourier Transform)calculation [10-11].The performance of signal acquisition method wasanalyzed using the real GPS IF data, which were collected bythe SAT-SURF receiver. The GPS receiver was stationary.The intermediate frequency is 4.152 MHz and the samplingfrequency is 16.3676 MHz. In our implementation, theconventional parallel in frequency output of the visiblesatellite with PRN-21 (Pseudo Random Noise code) is shownin Fig. 3.The quality of the results showed in the Parallel inFrequency (PiF) approach is proportional to the quantity ofpoints used to compute the Fast Fourier Transform (FFT).Another interesting point is the fact that the PiF gives theresults in the intermediate frequency range, so in order to getthe value of the Doppler it is necessary to subtract theobtained values by the intermediate frequency of the signal, oruse other kind of approach to bring down the signal to the baseband [11]. After performing the acquisition, control is handedover to the tracking loops, which are used to refine thefrequency and code phase parameters. The main purpose oftracking is to refine the carrier frequency and code phaseparameters, keep track, and demodulate the navigation data[12]. The values in Table 2 are passed into tracking loop sothey can keep track and demodulate the navigation datacorrectly.28 P a g ewww.ijacsa.thesai.org
(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 6, No. 3, 2015A combination of code tracking loop and carrier trackingloop is used in tracking procedure. In order to extractinformation from the incoming signals, GPS receivers trackthem by replicating the PRN code and adjusting its code delayand carrier phase continuously so as to guaranteesynchronization with the incoming signal. In Fig.4 a basiccode tracking loop is shown. The code tracking loop is to keeptrack of the code phase of a specific code. The code trackingloop uses a delay lock loop called an early-late tracking loop[13]. Integrate and Dump (I&D) are blocks that accumulatethe correlators outputs, and provide their In-phase I andQuadrature Q components.EIncomingsignalIPLLocal oscillatorIntegrate& DumpIEIntegrate& DumpIPIntegrate& DumpILPRN code generatorFig. 4. Basic code tracking loop block diagramAs it can be seen in Fig. 5, after 3 steps of the loop thealgorithm converges to the correct estimated delay. It isinteresting to notice that at this step the Early-Late becomeszero and the Prompt reaches its maximum value.Fig. 5. Modulus of Early, Late and Prompt correlationsIV.Fig. 3. The correlation results of frequency parallel-search method for PRN21 satelliteTABLE II.RESULTS FROM PIF GPS SATELLITES IN THE RECEIVED SIGNALPRNFrequency(Hz)Doppler (Hz)Code offset94.142e 006-696.915469124.139e 0061181.813756174.158e 006-2878.815194254.150e 0062636.44025274.149e 006-1303.011469304.144e 0063000.015955ADAPTIVE POSITIONING ALGORITHM IMPLEMENTATIONThis section is dedicated to algorithm implementation. Wepropose an innovative adaptive PVT algorithm compiled inMatlab environment. The specific computation flow diagramof our positioning algorithm is shown in Fig.6. Initially it isimportant to extract from the collected data the coordinates ofsatellites. Since we implement an Iterative Least Squares(ILS) algorithm the method of solving for GPS user’s positionis to linearize the pseudorange equations and calculate the userposition iteratively, starting with a user provided initialposition guess [14]. The next step is the calculation of thepseudoranges between satellites and user’s position. Thealgorithm computes the differences between the observed andpredicted ranges and gives as output the line-of-sight unitvectors from which it builds the geometry matrix. Theconvergence of the iterative solution will depend on thegeometry of the receiver-satellites system, which, in turn,affects the rank of geometry matrix H. Problems can occur ifH is rank-deficient or close to it, which can occur when all thesatellites lie in or very close to the same plane in threedimensional space.29 P a g ewww.ijacsa.thesai.org
(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 6, No. 3, 2015We obtain a least squares optimization only when thesolution is over-determined (i.e., number of satellites in viewgreater than four). When there are only the minimum fourmeasurements, the result is the solution of a set of linearStartDetermineSatellite Positionsequations. Regardless, solving these equations give uscorrections for our initial guess, which can now be reappliedto the initial guess, and the whole process is repeated until thecorrections become smaller than a threshold value [15].Initialize the user positionas the center of the EarthEstimatePseudorangesObserved MinusPredicted RangesLine of Sight UnitVectorsUpdate PositionEstimateSolve forCorrectionsCompute theGeometry MatrixNOCorrection ToleranceYESSave CurrentEstimateFinishApply Ionosphericand TroposphericCorrectionsFig. 6. Computational flow diagram of our positioning algorithmThere are different possibilities of implementing apositioning algorithm [14]. In our approach, data structuresare used as a faster and easier way to access the data neededfor position computation. In our Matlab implementation isdefined the “True Position” only for future comparison of theestimated positions obtained by our positioning algorithm andthe true one, in order to graphically depict the precision andaccuracy of the estimated positions. In order to evaluate theuser position, a linearization scenario is implemented bychoosing a linearization point, as our known referenceposition. Initially the linearization point is set the center ofEarth in ECEF (Earth-Centered Earth-Fixed) coordinatesystem with coordinates Lp [0; 0; 0; 0]. The linearizationpoint will be updated after each TOW (Time of Week)iteration, until in the end of the iteration to become theevaluated user position.All the parameters that will be used in the algorithm areinitially set to zero (initialization process). The chosenlinearization point is not a good approximation point becauseit is very far from our “True Position”, but it is suitable for theCold Start of the receiver (state that the receiver has noinformation of its position).During our measurements we collected a large amount ofdata for a total of 2340 TOW-s. The positioning algorithm istested for different number of iterations and the resultsobtained for the user position were approximately the same.This is due to the long observation time and due to the fact thatthe minimum number of fixed satellites were 6 (enough toproperly estimate user’s position). In Matlab environment,simulation time is not a crucial issue but in real receivers, timeis a very important constraint.The Navigation Solution in a first-order approximation isgiven by the following code lines:SatP(i,:) [xs ys zs];rho hat(i) norm(SatP(i,:) - Lp);a(i,:) (SatP(i,:) - Lp)/rho hat(i);The first one addresses the satellite coordinate’s triplet (Xs,Ys, Zs) , which are used in the second line in order to evaluatethe geometrical distance between the satellite position and thelinearization point Lp. After this process the a coefficients ofthe geometric matrix are written and it is important to say thatfor the first iteration and for the first TOW, we assume that thesatellite clock and user clock are synchronized. This happensonly for the first TOW, because the coordinates of the updatedlinearization point will be used as input for the successiveTOW.Since the strength quality of the signal is defined by theCarrier-to-Noise density Ratio, which is the ratio of the powerlevel of the signal in 1 Hz bandwidth, it is important toproperly weight satellites with low values of C/N0. Elevationangles and C/N0 values, as recorded by the receiver, are usedto model the pseudorange observations noise variance. Thechoice of the weight matrix is optimal when it equals theinverse of the variance-covariance matrix of the observations[16]. We implemented the model in [17] which uses the C/N0values of the GPS signals to estimate weights for least squareadjustment. Using this approach we achieved an improvementon the position estimation mostly in the vertical component.V.IONOSPHERIC AND TROPOSPHERIC CORRECTIONMODELSThe focus of this section is to evaluate the ionospheric andtropospheric effect on GPS positioning solution. Thepseudoranges are affected by errors, which can be modeled asGaussian random variables, with zero mean, independent and30 P a g ewww.ijacsa.thesai.org
(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 6, No. 3, 2015identically distributed, with variance[15]. The errorsaffecting the pseudoranges can be expressed by (1). ()()()(1)Equation (4) represents the total Tropospheric error[ ( )] andcontribution where[ ( )]. The humidity ratio in % in dry and wetconditions is given by (5):andWhereis the sum of Ionospheric andTropospheric error contributions, respectively. These twotypes of corrections are described in details in the followingsubsections.A. Ionospheric CorrectionsIonospheric corrections are implemented based on theKlobuchar model [3] which uses as input the parametersshown in Table 3. We designed a function ionogen.m tocalculate the delay caused by Ionosphere layer, which wascalled in our main PVT algorithm. Two are the main inputs ofthe ionospheric correction function. The first one is PERwhich is the period of the cosine function and implicates theinterval of the ionospheric activity in daytime. It is expressedby (2), whose inputs are taken from the ionosphere log file.TABLE III.(3)The inputs of the Klobuchar model were taken by loadingthe Elevation and Azimuth angles for each TOW and numberof fixed satellites. We observed that these coefficients areconstant even for different TOW (Fig. 2) and this result is dueto the fact that ionospheric parameters do not change in a shortmeasurement time.)(5)INPUT PARAMETERS OF KLOBUCHAR CORRECTION MODELReceiver generated terms uUser Geodetic Latitude WGS 84 (semi – circles) uUser Geodetic Longitude WGS 84 (semi – circles)EAGPStime n nElevation angle between the user and the satellite, measureclockwise positive from the true north (semi- circles)Geodetic azimuth angle of the satelliteReceiver’s computed system timeSatellite transmitted termsCoefficients of a cubic equation representing the amplitude ofthe delayCoefficients of a cubic equation representing the period (PER) ofthe model(2)Whereis the geomagnetic latitude of the Earth’sprojection of the ionospheric intersection point (meanionospheric height assumed to be 350 km). The second input isthe amplitude of the model given by (3).(VI.ANALYSIS OF THE RESULTSIn this section are shown the results of our work. Thereference frame used is the ECEF Cartesian coordinate system.In Fig. 7 the time evolution of Geometrical Dilution OfPrecision (GDOP) and the number of satellites are shown. Weobserve in table 4 that for all the TOW-s taken intoconsideration, the minimum number of fixed satellites is sixwhich is enough to properly estimate the user position becauseare required at least four satellites. When the number of fixedsatellites decreases, we observe increased values of GDOP, forinstance when the number of fixed satellites goes from 13 to 6the value of GDOP is increased from 1.59 to 5.29. When thenumber of fixed satellites increases, so more satellites come inview, the proper values of GDOP decrease because a betterestimation of the receiver’s position is achieved.B. Tropospheric CorrectionsThe signal refraction in the troposphere is separated intotwo components: the dry and the wet component, where thedry component contributes about 90 % of the totaltropospheric delay. The tropospheric delay is approximated byusing the Hopfield model [6], whose inputs in our algorithmare: T - Temperature in C.P - Pressure in hPa.Hu - humidity ratio in %.R - Earth radius: R 6371 km.- Satellite Elevation angle.This model is based on the relationship between the dryrefractivity at height h to the surface of Earth. We designed afunction in Matlab named tropogen.m to calculate the delaycaused by the Troposphere layer, as a function of elevationangle represented by the following equations0()()()(4)Fig. 7. The change of GDOP values and of number of satellites over TOW31 P a g ewww.ijacsa.thesai.org
(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 6, No. 3, 2015TABLE IV.GDOP VALUES AND NUMBER OF SATELLITES IN VIEW FORALL TOWValueMinimumMeanMaximumGDOP [m]1.592.4226.49Number of Satellites6714RealPositionEvaluatedPositionFig. 10. Estimated user’s position from the adaptive PVT algorithmTABLE V.SUMMARY OF DIFFERENT TRIALS COMPUTED FOR THE PVTSOLUTIONUser’s PositionWithout correctionIonosphere correctionTroposphere correctionIono Tropo correctionFig. 8. The true and estimated position in Geographic coordinatesEvaluatedPositionRealPositionFig. 9. Estimated position for the first 10 Times Of WeekIn Fig. 8 are plotted the true position of the receiver andthe cloud of points which represents the estimated position asan output of the positioning algorithm.After running the positioning algorithm with the raw dataof the first 10 Times Of Week the obtained estimated position(Fig. 9) has the following coordinates: Latitude 41.4540 andLongitude 19.6260 which is far from the true position. Thisis due to the linearization point which at the beginning is Lpand to the fact that the receiver is set up for the first time (coldstart). The receiver in cold start mode has no clue where itsposition might be and the first linearization point is far fromthe true ongitude19.8215019.8215019.8215019.82150Height (m)61.13418.52159.23114.275In Fig. 10 is shown the estimated position of the user afterthe iterations for all Times of Week and it has thesecoordinates: Latitude 41.31650 and Longitude 19.82150which are close to the true position (Latitude 41.31690 andLongitude 19.82150). The satellite-user geometry can have alarge impact on the accuracy of the PVT estimates obtainedfrom GPS. In other words, some satellite-user geometries willresult in a higher accuracy solution than others. As such, it isuseful to have a way of comparing different satellite-usergeometries. The metric normally used for measuring thisimpact is dilution of precision (DOP), which represents thedegree to which satellite-user geometry dilutes the accuracy ofthe PVT. DOPs can be viewed as the link between thepseudorange errors and PVT estimation errors. Since DOPschange as the user-satellite geometry changes over time asillustrated in Fig.7, this implies that a given level ofpseudorange measurement error will translate into differentlevels of PVT errors.After applying the ionospheric and tropospheric correctionmodels, the error in the vertical component (height z) issignificantly reduced. Figure 8 shows the estimated positionsand the true position in Geographical coordinates for a betterunderstanding of the atmospheric residual errors. TheKlobuchar model reduces the vertical error with a value equalto 42.6 m. The Tropospheric Hopfield model applied in ouradaptive PVT algorithm, gives a slight correction to thevertical error in the amount of 1.9 m. This was an expectedoutcome because Tropospheric error’s impact is lowercompared to the Ionospheric one, in the total errorcontribution. These important results are summarized in Table5.32 P a g ewww.ijacsa.thesai.org
(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 6, No. 3, 2015Finally, the user’s position estimated by our adaptive ILSpositioning algorithm for all GPS epochs or TOWs is:[41.31650 N; 19.82150 E; 14.275 m]. This estimated position isvery close to the true position which is illustrated in Fig. 10.VII. CONCLUSION AND FUTURE WORK[2][3][4][6][7]The aim of this paper was to evaluate the positioningperformance of a single-frequency software GPS receiverusing Ionosperic and Tropospheric corrections. We proposedan adaptive ILS algorithm, where we integrated Klobuchar andHopfield mathematical correction models, enabling data postprocessing. In our measurement process we used the SATSURF receiver. In order to minimize the impact of large errorsin the position estimation, we applied the Weighted Matrix. Inthe first ten TOW-s test we obtained very bad results in theuser position estimation, this was due to cold start of thereceiver (where the receiver has no clue about its position) andbecause the initial linearization point was chosen to be very farfrom the user’s True Position. Since the goal of a positioningalgorithm is to provide the user position in a minimum numberof iterations we show that three iterations were enough tofulfill this requirement. The final user position obtained by ourpositioning algorithm was 41.31650 North, 19.82150 East,61.134 m Up. Applying the Klobuchar model for Ionosphericcorrection, a reduction by 42.6 m of the vertical error wasachieved; however this model did not affect significantly thehorizontal positioning. On the other hand, the integration ofHopfield Tropospheric model in our positioning algorithm,gave a slight improvement of the vertical error by 4.25 mcompared to ionospheric correction. This is a good result,taking into account that our receiver is a mass market receiverworking in single frequency. In our future work we will focuson the mitigation of other error’s contribution such asrelativistic, ephemerides and satellite clock errors. We willalso investigate the positioning performance achieved after theapplication of EGNOS and differential corrections, usingdouble frequency GPS receivers for Precise Point Positioningapplications.[1][5]REFERENCESR. Warnant, K. Ivan, P. Marinov, M. Bavier, and S. Lejeune,“Ionospheric and geomagnetic conditions during periods of degradedGPS position accuracy: 2.RTK events during disturbed and quietgeomagnetic conditions”, Advances in Space Research,Vol. 39, No. 5.,pp.881-888, 2007.G . Hochegger, B. Nava, S.M. Radicella and R. Leitinger “A family ofionospheric models for different uses”, Phys. Chem. Earth, 25 (4), 307310, 2000.J.A. Klobuchar, “Ionospheric time-delay algorithm for single-frequencyGPS users”, IEEE Trans. Aerosp. Electron. Syst., AES-23 (3), 325-331,1987.B. Hofmann-Wellenhof, H. Lichtenegger, J. Collins, “Global PositioningSystem:Theory and Practice”, 5th revised edition, Springer-Verlag, 382pp. 2001.[8][9][10][11][12][13][14][15][16][17]R.B. Langley, “Propagation of the GPS Signals”, In: Kleusberg, A. andTeunissen, P.J.G. (eds), GPS for Geodesy (2nd edition), SpringerVerlag, Berlin Heidelberg New York, 111-150, 1998.H.S. Hopfield, “Two–quadratic tropospheric refractivity profile forcorrection satellite data”, Journal of Geophysical Research, 74(18), 4487– 4499, 1969.P. X. Quang, L. Lo Presti, F. Dominici, G. Marucco, “SAT-SURF andSAT-SURFER: a flexible platform for both R&D and training onGNSS”. In: 2nd GNSS Vulnerabilities and Solutions 2009 Conference,Baska, Krk Island, Croatia, 2-5 September, 2009. pp. 1-12.J.L. Leva, "Relationship between navigation vertical error, VDOP, andpseudo-range error in GPS", Aerospace and Electronic Systems, IEEETransactions, vol.30, no.4, pp.1138,1142, Oct 1994.L. Dong, Ch. Ma, and G. Lachapelle, "Implementation and Verificationof a Software-Based IF GPS Signal Simulator," Proceedings of the 2004National Technical Meeting of The Institute of Navigation, San Diego,CA, January 2004, pp. 378-389.J. Tian and L. Yang, "A Novel GNSS Weak Signal Acquisition UsingWavelet Denoising Method," Proceedings of the 2008 NationalTechnical Meeting of The Institute of Navigation, San Diego, CA,January 2008, pp. 303-309.B. Y. T. James, “Fundamentals of global positioning system receivers asoftware approach”, A John Wiley&sons, New York, 2004.P. Lian, G. Lachapelle, and Ch. Ma, "Improving Tracking Performanceof PLL in High Dynamics Applications," Proceedings of the 2005National Technical Meeting of The Institute of Navigation, San Diego,CA, January 2005, pp. 1042-1052.R. Peter and B. Nicolaj, “Design of a single frequency GPS softwarereceiver”, Aalborg University, pp. 31-35, 2004.W. Li, Z. Yuan, B. Chen, and W. Zhao, "Performance comparison ofpositioning algorithms for complex GPS systems", DistributedComputing Systems Workshops, 32nd International Conference onDistributed Computing Systems (ICDCSW), pp.273-278, Macau,China,18-21 June 2012.P. Misra, and P. Enge, “Global Positioning System: Signals,Measurements and Performance, Revised Second Edition”, Lincoln,MA: Ganga-Jamuna Press, 2011, ISBN 0-9709544-1-7.C.C.J.M. Tiberius, “The GPS data weight matrix: what are the issues?”,Proceedings of the 1999 National Technical Meeting of The Institute ofNavigation, January 25 - 27, 1999, Catamaran Resort Hotel, San Diego,CA, pp. 219 – 227.H. Hartinger and F. Brunner, “Variances of gps phase observations :Thesigma-e model,” GPS Solutions, vol. 2, pp. 35–43, 1999.AUTHOR PROFILEAlban RAKIPI has graduated the Faculty ofInformation Technology, Polytechnic University of Tiranain 2009. He holds a Master of Sciences diploma inTelecommunication Engineering from 2011 and a secondlevel specializing master diploma in “Navigation andRelated Applications” from Polytechnic University ofTurin, Italy. Currently he is a full-time Lecturer within theDepartment of Electronics and Telecommunications at Faculty of InformationTechnology, Polytechnic University of Tirana.His work focuses on signal processing, satellite
measurements I. INTRODUCTION Hopfield Model [Recently, there is an increase interest in positioning techniques based on GNSS (Global Navigation Satellite Systems) such as GPS (Global Positioning System). GPS is a satellite-based navigation radio system which is used to ve
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