A Hidden Markov Model for Pedestrian NavigationJochen Seitz1, Thorsten Vaupel2, Steffen Meyer2, Javier Gutiérrez Boronat2 and Jörn Thielecke1

1Friedrich-Alexander University of Erlangen-Nuremberg2Fraunhofer Institute for Integrated Circuits IIS

Abstract—We present an algorithm for pedestrian navigationoptimized for smart mobile platforms using the present low-costsensors and the limited processing power. The algorithm is basedon a Hidden Markov Model that combines Wi-Fi positioningand dead reckoning. The hidden states are the positions of theWi-Fi fingerprints in the database. The state transition includesdead reckoning based on step length estimation from accelerationmeasurements and compass heading calculated from magneticfield measurements. In the measurement update a databasecorrelation of the actual Wi-Fi signal strength measurementswith the stored values in the fingerprints has been performed. Insimulations and tests we demonstrate that in this way ambiguitiescommon in Wi-Fi positioning can be reduced. Therefor, higheraccuracy and robustness can be achieved.

Index Terms—Dead Reckoning, Hidden Markov Models, Pe-destrian Navigation, Positioning, Wireless LAN

I. INTRODUCTION

The use of Hidden Markov Models (HMM) to fuse differentsensor outputs has been deployed successfully for autonomousmobile robots, for example in [4] and [10]. Frequently usedsensors are laser range scanners, ultrasonic sensors and odome-ters in combination with powerful computer platforms anddetailed maps of the surrounding area. If high precision isneeded, the sensor equipment is expensive and bulky andthe computational costs are high. For pedestrian navigationthe setup has to be customer-friendly. Hence, the installedsensors must be cheap, small in size and preferably integratedin consumer devices like mobile phones. Commonly the firstchoice for navigation is the global positioning system (GPS),as it is installed in more and more portable smart phones.However, the lack of precision and availability of GPS in urbanand indoor environments is a prevalent problem in pedestriannavigation.

In this paper we focus on the fusion of low-cost accelero-meters, magnetic field sensors and wireless LAN (Wi-Fi R©[15]) modules in order to achieve robust positioning in citiesand indoors. These are integrated in many modern mobilephones together with GPS. Our test system is a HTC Herosmart phone with an AndroidTM [1] operating system.

Besides for communication, Wi-Fi modules can also beused for positioning [3]. Because of an increasing number ofpublic and private access points in urban environments, Wi-Fipositioning becomes more and more attractive for pedestriannavigation [7]. The so called Wi-Fi fingerprinting is done bycorrelating received signal strength (RSS) measurements withentries of a fingerprinting database. The database is createdby earlier taken RSS measurements and referenced with the

coordinates of the position, where they have been observed.Three-axial accelerometers can be used for step detection andstep length estimation, as presented in [6], and the three-axial magnetic field sensors to estimate the horizontal compassheading.

We combine the compass heading with the step length tocalculate the change of position relative to an initial position.This approach is known as dead reckoning. The problem indead reckoning is the small accumulating error with each stepthat causes an increasing absolute positioning error over time.While dead reckoning achieves high precision within shorttime periods, the positioning error of Wi-Fi positioning isnot increasing with time, but less short-term accurate with anabsolute error in the range of meters [7]. Therefore the sensordata fusion of dead reckoning and Wi-Fi positioning yields asynergetic effect resulting in higher robustness and precision.

In [13] and [16] particle filters are presented for the in-tegration of Wi-Fi positioning based on fingerprinting andinertial navigation systems. Particle filters suffer from highcomputational costs, depending on the number of particlescomputed. Kalman filters and extended Kalman filters areinappropriate for the sensor data fusion, as the assumptionof Gaussian measurement distributions is in conflict with theambiguous outputs of Wi-Fi fingerprinting algorithms. In [11]a HMM for probabilistic Wi-Fi positioning is presented andthe efficiency is demonstrated.

Because of the limited processing power of mobile phoneswe chose this approach. In this paper we extend the HMM forWi-Fi positioning of [11] by including dead reckoning into theposition estimation process. In this way ambiguities commonin Wi-Fi positioning can be reduced; higher robustness canbe achieved. In Section II the components and equations ofthe HMM are presented. The general performance, drawbacksand possibilities are evaluated in Section III by simulations.In Section IV experimental results in typical urban indoor andoutdoor environments are presented. Finally, conclusions aredrawn in Section V.

II. HIDDEN MARKOV MODEL

A Markov model is called hidden if it contains an un-derlying stochastic process that is not directly observable,but can be observed through another stochastic process [8].The proposed HMM is based on the discrete positions of theWi-Fi fingerprints as the hidden states and the received signalstrength (RSS) Wi-Fi measurements as the observable states.The iterative estimation process with the HMM is presentedin Fig. 1. At first an initial probability distribution p0 for

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Fig. 1. Iterative estimation process with the hidden Markov model

the user position has to be provided. If there is no initialinformation, this can be assumed by an uniform distribution.In the movement update the state transition, the change ofposition, is based on dead reckoning. In the RSS update adatabase correlation is performed that matches the currentRSS sensor measurements with the stored RSS values ofthe fingerprints in the database. After the RSS update it isnecessary to normalize the probabilities for all positions. Fromdead reckoning the change in position can be calculated andfrom the RSS measurements correlation coefficients for thefingerprints. The position of the pedestrian is not directlyobservable.

A. Movement Update

Assume we have already calculated the probability distribu-tion p(Lt−1|Mt−1,Rt−1) for the location Lt−1 at time t− 1given all measurements up to and including time t−1.Mt−1

denotes the set of movement measurements and Rt−1 the setof RSS measurements. In the movement update we want tocalculate the probability distribution p(Lt−1|Mt,Rt−1). Thechange in position is captured by dead reckoning. We estimatethe step length from the acceleration measurements using analgorithm for pedometers from [14] and the enhancementof [9]. For the heading of the user we assume that it isidentical to the orientation of the mobile device, measured bya magnetic compass. The movement update is based on theMarkov assumption that the user position Lt at time t dependsonly on the previous position Lt−1 and the latest distance andangle measurements Dt and Φt indicating the movement ofthe user:

p(Lt|Mt,Rt−1) =∑l′

p(Lt|Lt−1 = l′, Dt,Φt)

· p(Lt−1 = l′|Mt−1,Rt−1). (1)

The movement update for each user position Lt = l is thesum over all probabilites of previous positions l′ weighted bythe transition probability p(Lt|Lt−1 = l′, Dt,Φt), given thedistance and angle measurement. For this transition probabilitywe can write using Bayes law:

p(Lt|Lt−1, Dt,Φt) =p(Dt,Φt|Lt, Lt−1) p(Lt|Lt−1)

p(Dt,Φt|Lt−1). (2)

Given the distance and angle measurements are always inde-pendent of each other, we obtain:

p(Lt|Lt−1, Dt,Φt) =

p(Dt|Lt, Lt−1) p(Φt|Lt, Lt−1) p(Lt|Lt−1)p(Dt) p(Φt)

. (3)

Here we have assumed in the denominator that the distanceand angle measurements are independent of the position Lt−1,where we started the movement measurement. By the transi-tion probability p(Lt|Lt−1) of the numerator, we could takeinto account, e. g., that a movement to neighboring locationsis more likely than to very distant locations. However, bythe movement measurements Dt and Φt we have already areliable measure for what locations are more likely than others.Therefore we assume for simplicity that p(Lt = l|Lt−1 = l′)is a constant for all pairs (l, l′). Inserting now Eq. (3) intoEq. (1) we finally obtain for the movement update:

p(Lt|Mt,Rt−1) =

c ·∑l′

p(Dt|Lt, Lt−1 = l′) p(Φt|Lt, Lt−1 = l′)

· p(Lt−1 = l′|Mt−1,Rt−1). (4)

Here we have collected all probabilities independent of l′ in theconstant c. This constant can be readily determined at the endof the movement update by normalization, since the calculateddistribution is a probability distribution, i. e., summing over allpossible locations Lt = l must yield 1.

The probability distributions p(Dt|Lt, Lt−1) andp(Φt|Lt, Lt−1) characterizing the distance and anglemeasurement are assumed to be Gaussian distributions:

p(Dt = dt|Lt = l, Lt−1 = l′) =1√

2π · σd

·e− 1

2σ2d

(dt−d(l,l′))2

,

(5)

p(Φt = ϕt|Lt = l, Lt−1 = l′) =1√

2π · σϕ

·e− 1

2σ2ϕ

(ϕt−ϕ(l,l′))2

.

(6)Implicitly, we have assumed here that all measurements aretaken just at the moment when the new reference point l isreached. Fig. 2a and 2b visualize these conditional probabili-ties for the case that the previous position l′ is at the centerof the coordinate system. Multiplying both conditional proba-bilites yields the probability distribution of Fig. 2c. Obviously,combining the distance and angle measurement gives a ratherprecise indication whereto the position changes.

B. RSS Update

In the RSS update we calculate the probability distri-bution for the location Lt given all movements and RSSmeasurements up to and including time t by weighting theprobability of the movement update from (1) by the RSS Wi-Fimeasurement vector rt that has been collected at Lt:

p(Lt|Mt,Rt) = p(rt|Lt) · p(Lt|Mt,Rt−1). (7)

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(a) (b) (c)

Fig. 2. Transition probability distributions in gray scale, white equatesprobability density 0, for (a) step length, (b) compass-heading and (c) thecombination of both, with one estimated user position in the center

For p(rt|Lt) we use the probabilistic database correlationof [12]. A Gaussian distribution is assumed for each RSSmeasurement rt,i of access point i at a fingerprint location.So a standard deviation σi for each RSS measurement hasto be stored additionally in the database. Then we weight theactual RSS measurements rt,i by a probability density functionwith the database entry ri and the standard deviation σi. Asmost Wi-Fi devices provide discrete RSS values quantizedwithin an interval of approximately 1 dB, we integrate theGaussian probability distribution between a = rt,i − 0.5 dBand b = rt,i+0.5 dB. We repeat this procedure for all observedaccess points and multiply the results to obtain the probabilityfor the measurement vector rt at location Lt = l:

p(rt|Lt = l) =∏

i

1√2π · σi

·b∫

a

e− 1

2σ2i

(r−ri(l))2

dr. (8)

In Fig. 3 an example for the probabilistic database corre-lation is depicted. During database creation the same accesspoint could be observed at both fingerprints. The mean RSSvalue of −40 dBm at fingerprint 1 is higher than the −60 dBmat fingerprint 2. The reason is, that fingerprint 1 is closer tothe access point or less affected by signal shadowing. Further-more the standard deviation σi of fingerprint 2 is higher, assuggested in [3]. So the probability that measurement rt,i wastaken at fingerprint 1 is much higher.

(a) (b)

Fig. 3. Example for calculating the probability of a measured signal strengthrt,i = −45 dBm for (a) fingerprint 1 and (b) fingerprint 2 for the same accesspoint i using probabilistic database correlation

After the RSS update has been executed the resultingprobabilities of the user position have to be normalized. Thefinal user position [x, y] can be estimated by calculatingthe weighted mean of all possible positions l, for example

weighted by their probabilities p(Lt = l|Mt,Rt) from eq.(7). The advantage of the proposed HMM is that the databasecorrelation is directly included in the fusion process. Ambi-guities in the user position caused by similar fingerprints canbe handled in the RSS update.

III. SIMULATIONS

To validate our models we simulated a typical Wi-Fipositioning scenario including signal shadowing. The signalstrength for the fingerprinting database was calculated bymodeling the path loss together with log-normal fading asdepicted in Fig. 4. Furthermore we added white Gaussiannoise to the ideal step length, compass heading and RSSmeasurements. The grid spacing in a 50 m long and wide areawas 1 m. We placed nine access points in the area, one in each

(a) (b)

Fig. 4. Path loss (a), with log-normal fading (b) for one access point, placedin the center

corner, one in the middle and one in the middle of each outeredge. For each access point we applied the path loss combinedwith log-normal fading. The resulting signal strength valuesare stored for every equidistant grid point in the fingerprintingdatabase. White noise has been added to the RSS values inthe database with a standard deviation of 6 dB, which is highbut in the typical range from 2 dB to 10 dB depending on theWi-Fi device and drivers [5]. Finally to simulate discrete RSSmeasurements the values have been rounded.

To emulate the step length and compass measurements weimplemented a random path with 10 000 steps in the area.We added white Gaussian noise with a standard deviation of0.3 m for the estimated step length and a standard deviationof 10◦ for the compass. The step frequency of pedestriansranges usually from 0.5 Hz to 3 Hz and the sampling rate ofWi-Fi devices is usually some seconds. As the step frequencyof pedestrians is higher in most cases than the sampling rateof the Wi-Fi device, the movement update is repeated witheach step until the next Wi-Fi measurements are available.

To obtain an impression Fig. 5 presents four steps of theiterative workflow. It starts with the probability distribution ofthe user position after a RSS update Fig. 5(a,b). Then, in thesimulation, the user walks 12 steps, about 10 m, in x-direction.The result of the movement updates can be seen in Fig. 5(c,d).The probability distribution slides in x-direction. Furthermoreit is broadened and flattened because of step length andcompass heading uncertainties. After the 12 steps new Wi-Fi

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(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 5. Example steps of the iterative HMM from simulation results in side(left) and top view (right): RSS update 1 with normalization (a,b), movementupdate with 12 steps in x-direction (c,d), Wi-Fi database correlation results(e,f) and RSS update 2 with normalization (g,h)

measurements are available. The results of the probabilisticdatabase correlation are presented in Fig. 5(e,f). In this casetypical ambiguities appear for the user position. The corre-lation results are used in the RSS update in Fig. 5(g,h) andweighted by the results of the movement update Fig. 5(c,d).As displayed in Fig. 5(g,h) the ambiguities from the databasecorrelation can be resolved.

To evaluate the benefit of including dead-reckoning in theHMM the probability distributions for the user positions werecalculated using different combinations of the sensors. Wetested the interaction of the elements described in SectionII and the improvements in accuracy compared to just Wi-Fifingerprinting.

Simulation results for the position estimation with theproposed HMM are shown in Fig. 6. The positioning errors and

the 70, 90 and 95 percentiles are presented. The simulation re-sults show improvements in accuracy by using dead reckoningin the movement update. Using only step detection with a fixed(dist.) or an estimated step length (step) the positioning error isreduced by more than 50 %. Furthermore, with the additionaluse of compass heading (comp.) it is reduced by more than75 %. The improvement using step length estimation insteadof a fixed step length is very low. The reason is that after themovement update the user position is matched to the discretepositions of the fingerprints. To improve the effect of the steplength estimation between the fingerprints a lower distancecould be used, but the problems are that the computationalcosts grow with the amount of fingerprints and, with lowerdistances in between, the fingerprints differ less and less insignal space, as investigated in [11].

Fig. 6. Simulation results for: stand-alone Wi-Fi fingerprinting (FP) anddifferent combinations of the probabilistic database correlation together witha fixed step length (dist.), step length estimation without (step) and withcompass heading(comp.)

But low-cost magnetic compasses in hand-held devices havedifferent noise levels. Furthermore, magnetic disturbances,caused for example by building structures, distort the compassheading. Hence, we tested the position estimation using steplength estimation and applying various noise levels of thecompass heading. As shown in Fig. 7 there is an increaseof the positioning errors until a standard deviation of about90◦. Compared to Fig. 6, with standard deviations higher than90◦, the error remains on the same level as with step lengthestimation. The standard deviation of the magnetic compasshas an influence on the positioning error, but already withstandard deviations lower than 90◦ there is an improvement.

IV. EXPERIMENTAL RESULTS

In this section we will compare the results from our sim-ulations with the results from two experimental setups. Wecollected acceleration, magnetic compass and Wi-Fi measure-ments with a HTC Hero mobile phone based on the Androidoperating system. The acceleration sensors and the magneticcompass are integrated on one chip. If both are used at thesame time, the maximum sampling rates are approximately30 Hz for the acceleration measurements and 20 Hz for thecompass. The sampling rate for the Wi-Fi measurements wasabout 5 Hz. For position estimation we included the HMMin the navigation framework of the Wi-Fi positioning system

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Fig. 7. Mean positioning errors and percentiles using various standarddeviations for the compass heading

awilocTM [2] of the Fraunhofer Institute for Integrated CircuitsIIS.

Our first scenario is the historic square in the center ofNuremberg, the famous Christmas market. The second sce-nario is the building of the Fraunhofer IIS in Nuremberg. Itis a typical office building. There we collected measurementsindoors and outdoors.

A. Market Square

The measurements were taken after the market was set-up,but before it was started. No visitors were on the market.Several persons collected the measurements with the presentedsensing device. During the measurements they walked onpreviously planned paths between the stalls on the market intwo directions. In Fig. 8 the tracks can be identified by thecircles representing the collected fingerprints. In the databasethe identifiers of the received access points together withtheir RSS measurements are stored per fingerprint. In theRSS update we used an approximation of Eq. (8) with afixed standard deviation for the RSS measurements similarto the one presented in [3]. This was necessary because nostandard deviations for the RSS measurements were stored inthe database.

(a) (b)

Fig. 8. Fingerprints collected at the Christmas market (a) and in detail theupper right corner (b); the numbers indicate the number of received accesspoints

After we established the database we collected measure-ments for position estimation along the path shown in Fig. 9.The estimated positions by stand-alone Wi-Fi fingerprintingand by applying the HMM using step length estimation andcompass heading in the movement update are presented.Because of the lack of shadowing objects, like e. g. walls in

(a)

(b)

Fig. 9. Reference path (solid line) and estimated positions (crosses) of (a)stand-alone Wi-Fi positioning and (b) the HMM on the Christmas market

buildings, the market square is not an ideal location for Wi-Fipositioning. This causes a lot of ambiguities in the databasecorrelation and many result in large distances between twosequently estimated positions. We call this a hopping ofthe position. The result is the cloud-like arrangement of theestimated positions in Fig. 9a. The resulting mean positioningerror and the 70, 90 and 95 percentiles are presented in Fig. 10.

On the one hand, there are less ambiguities in the case ofthe HMM, so the estimated positions are closer to the walked

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Fig. 10. Mean positioning errors and percentiles for the position estimationusing stand-alone Wi-Fi positioning and the HMM for the measurementscollected on the Christams market

path, as can be seen in Fig. 9b. On the other hand the positionsare not distributed equally along the path and the improvementin accuracy is much lower than estimated in the simulations.The reason are falsely detected steps. Two factors complicatestep detection and step length estimation: the low samplingrate of 30 Hz and the attenuation in the acceleration signalcaused by carrying the device in the hand. The second reasonthat reduces accuracy is the difference between user headingand the orientation of the sensing device that forced us touse a high standard deviation of 45◦ for the compass in themovement update.

B. Office Building

After testing the system on the Christmas market we chosea surrounding with better conditions for Wi-Fi positioning. Wepicked the office building of the Fraunhofer IIS in Nuremberg.In this case we can use our own Wi-Fi infrastructure incombination with the signals received from access pointsplaced in other buildings and companies located nearby. The

Fig. 11. Architectural drawing of the office building and its surroundingswith the walked path during RSS data collection indoor (thin dotted line) andoutdoor (dashed line)

building structure causes signal shadowing, so less ambiguities

result in the database correlation process. The equipment andthe proceedings have been the same as in the first scenario. InFig. 11 an architectural drawing of the office building and itssurroundings is presented together with the walked path duringdata collection. The user started indoor on the first floor walkedthe depicted indoor path along the corridors, down the stairsto the ground floor and out of the building. He circled thebuilding two times in different distances to the building. Thenhe entered the building again and walked the same indoor pathas on the first floor now on the ground floor.

(a)

(b)

Fig. 12. Reference path (solid line) and estimated positions (crosses) of (a)Wi-Fi positioning stand-alone and (b) the HMM at the office building

In Fig. 12 the positioning results are presented. As expectedthe estimated positions by Wi-Fi fingerprinting are less scat-tered around the reference path than on the market square.

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Again with the use of dead reckoning in the HMM, ambiguitiescan be handled. Hence, the estimated positions are mainly onthe path. In this case we paid special attention to step detectionby carrying the sensing device in one hand horizontally infront of the body to obtain optimal acceleration measurements.Because of this, almost all steps can be detected. So inthis scenario the estimated user positions are more regularlydistributed on the path, as can be seen in Fig. 12b. The use ofthe compass still caused difficulties. On the one hand, the userheading was better aligned to the orientation of the sensingdevice, but on the other hand the ferromagnetic materials in thebuilding structure caused larger magnetic disturbances. Hence,we still had to use 45◦ as standard deviation of the compassmeasurements in the movement update.

(a)

(b) (c)

Fig. 13. Mean positioning errors and percentiles for the position estimationusing stand-alone Wi-Fi positioning and the HMM for the measurementscollected at the office building for (a) the whole path, (b) the indoor partand (c) the outdoor part of the path

The mean positioning error and the 70, 90 and 95 percentilesare presented in Fig. 13 for the indoor and the outdoor part ofthe path. In Fig. 14 the corresponding distributions of the po-sitioning error along the path are depicted. One measurementpack consists of one RSS measurement per received accesspoint and several acceleration and compass measurementscollected during the RSS measurements. The first 800 and thelast 300 measurement packs have been collected indoors andthe ones in between outdoors. The accuracy indoors is signif-icantly higher. The Wi-Fi positioning error is indoors aboutone third of the one outdoors. In summary, Wi-Fi positioningbenefits especially outdoors from using dead reckoning in themovement update of the HMM.

V. CONCLUSIONS

A method to fuse the RSS data of low cost sensors onmobile phones for precise and robust calculation of posi-

(a)

(b)

Fig. 14. Distribution of the positioning errors of (a) stand-alone Wi-Fipositioning and (b) the HMM for the data collected in the indoor and outdoorenvironment at the office building and the mean positioning error (solid line)together with the 70 (dash/dotted line), 90 (dotted line) and 95 (dashed line)percentiles

tion and heading of pedestrians in urban environments wasproposed. The presented HMM can deal with ambiguitiesresulting from Wi-Fi fingerprinting. The algorithm is com-putationally efficient because of the discrete space model,based on the positions of Wi-Fi fingerprints. We included deadreckoning in the movement update of the HMM. In simulationswe evaluated the benefits of dead reckoning together withWi-Fi positioning. Depending on the standard deviation ofthe compass the simulation results promised a reduction ofpositioning errors up to 75 % compared to stand-alone Wi-Fipositioning. In two real scenarios involving a Christmas marketand an office building the proposed algorithm was tested withreal measurement data collected by mobile phones. IndoorsWi-Fi positioning in general works very well, because ofless ambiguities. Outdoors, it benefits especially from deadreckoning in the movement update of the HMM.

ACKNOWLEDGMENT

This work was funded by the Bavarian ministry of economicaffairs, infrastructure, transport and technology in the scopeof the strategic program "Bayern 2020". The authors want to

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especially thank Dipl.-Ing (FH) Stephan Haimerl for his helpconcerning the Wi-Fi positioning system awiloc [2] and Dipl.-Ing (FH) Jasper Jahn concerning step detection, step lengthestimation and dead reckoning. Both work for the FraunhoferInstitute for Integrated Circuits IIS in Nuremberg.

REFERENCES

[1] android, “http://www.android.com/,” android is a registered trademarkof Google Inc.

[2] awiloc, “http://www.awiloc.com/,” awiloc is a trademark of the Fraun-hofer Institute for Integrated Circuits IIS, Germany, 2010.

[3] P. Bahl and V. Padmanabhan, “Radar: an in-building rf-based userlocation and tracking system,” in INFOCOM 2000. Nineteenth AnnualJoint Conference of the IEEE Computer and Communications Societies.Proceedings. IEEE, vol. 2, 2000, p. 10.

[4] D. Fox, “Markov localization: A probabilistic framework for mobilerobot localization and navigation,” Ph.D. dissertation, Bonn, Germany,Institute of Computer Science III, University of Bonn, 1998.

[5] F. Kiefer, “Untersuchung des Hardwareeinusses bei WLAN-Lokalisie-rungssystemen,” Studienarbeit, Germany, department of electrical, e-lectronic and communication engineering, University of Erlangen-Nuremberg, 2008.

[6] A. Kuo, “A simple model of bipedal walking predicts the preferredspeed–step length relationship,” Journal of biomechanical engineering,vol. 123, p. 264, 2001.

[7] S. Meyer, T. Vaupel, and S. Haimerl, “Wi-fi coverage and propagationfor localization purposes in permanently changing urban areas,” in IADISInternational Conference Wireless Applications and Computing, 2008.

[8] L. Rabiner and B. Juang, “An introduction to hidden markov models,”ASSP Magazine, IEEE, vol. 3, no. 1, pp. 4 – 16, Jan 1986.

[9] J. Scarlett, “Enhancing the performance of pedometers using a singleaccelerometer,” Application Note AN-900, Analog Devices, 2007.

[10] S. Thrun, W. Burgard, and D. Fox, Probabilistic Robotics. The MITPress, Cambridge, Massachusetts, USA, 2005.

[11] M. Wallbaum, “Indoor geolocation using wireless local area networks,”Ph.D. dissertation, Chair of Computer Science 4, Department of Com-puter Science, RWTH Aachen University, 2005.

[12] M. Wallbaum and T. Wasch, “Markov localization of wireless localarea network clients,” in Lecture Notes in Computer Science, vol. 2928.Springer, 2004, pp. 1–15.

[13] H. Wang, A. Szabo, and J. Bamberger, “Performance comparison ofnonlinear filters for indoor wlan positioning,” in The 11th InternationalConference on Information Fusion, vol. 1, 2008, p. 7.

[14] H. Weinberg, “Using the adxl202 in pedometer and personal navigationapplications,” Application Note AN-602, Analog Devices, 2002.

[15] Wi-Fi, “http://www.wi-fi.org/,” Wi-Fi is a registered trademark of theWi-Fi Alliance, 2003.

[16] O. Woodman and R. Harle, “Pedestrian localisation for indoor en-vironments,” in Proceedings of the 10th international conference onUbiquitous computing. ACM New York, NY, USA, 2008, pp. 114–123.

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