IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 46, NO. 1, FEBRUARY 1997
`
`65
`
`Estimating Position and Velocity of
`Mobiles in a Cellular Radio Network
`
`Martin Hellebrandt, Rudolf Mathar, and Markus Scheibenbogen
`
`Abstract— Determining the position and velocity of mobiles
`is an important issue for hierarchical cellular networks since
`the efficient allocation of mobiles to large or microcells depends
`on its present velocity. In this paper, we suggest a method
`of tracing a mobile by evaluating subsequent signal-strength
`measurements to different base stations. The required data are
`available in the global system for mobile (GSM) system. The basic
`idea resembles multidimensional scaling (MDS), a well-recognized
`method in statistical data analysis. Furthermore, the raw data
`are smoothed by a linear regression setup that simultaneously
`yields an elegant, smoothed estimator of the mobile’s speed. The
`method is extensively tested for data generated by the simulation
`tool GOOSE.
`
`I. INTRODUCTION
`
`preferable that can be implemented within the existing GSM
`standard, independent of external information.
`Two quantities are near at hand to obtain distance and
`speed information: signal strengths of different base stations
`measured at a mobile and corresponding propagation times.
`Both parameters are subject to strong fluctuations caused by
`short-term fading, shadowing, and reflections such that a so-
`phisticated method is necessary to translate signal strength into
`distance information. Several procedures have been suggested
`in literature.
`First attempts to monitor the position of vehicles arose from
`the need for knowing the disposition and status of vehicles
`in transport systems (see [16] and [17] for an overview).
`Reference [3] describes a system where the signal strength
`of a mobile’s transmitter is measured on a statistical basis
`by a set of base stations. From a priori information of the
`corresponding contours, the most probable location of the
`mobile is determined. Concerning the basic idea, this approach
`is related to what we will develop in our paper. However, in [3]
`no feasible search procedure for realistic scenarios is offered.
`Using elementary geometric considerations and a least
`squares estimate to smooth measurement errors, [19] develops
`a trilateration method based on radio-frequency (RF) travel
`time measurements between the vehicle and fixed sensors
`located at the edges of a square. This method was further
`pursued by [4] for channel allocation in cellular networks.
`Refined trilateration techniques for road environments, using
`time delays between reception of a mobile’s transmission at
`different nodes as input data, are investigated by [21].
`Signal-strength measurements are used in [6] to assign a
`mobile to a certain base station coverage zone. A channel
`allocation algorithm is introduced that uses this information
`as a basic ingredient. In [2], the area of interest is divided into
`small sub areas that are (not uniquely) characterized by a list
`of discrete signal power values from different base stations.
`However, a nonsatisfying behavior of the system is reported
`if complicated shadowing environments are considered.
`Recently, in [9] and [10], adaptive schemes based on hid-
`den Markov models, neural networks, and pattern-recognition
`methods have been employed to estimate the position of
`mobiles.
`Some work has also been devoted to the estimation of
`velocity only. The elapsed time until a cell handoff in a
`picocell occurs yields a rough estimate of a mobile’s speed and
`forms the basis for a handover algorithm in [20]. Reference [8]
`uses the number of level crossings of the average signal level
`to estimate the velocity. Reference [12] develops a method
`0018–9545/97$10.00 ª
`
`RAPIDLY growing load is expected for cellular radio net-
`
`works in the near future. Since the number of channels,
`either physical, time division multiple access (TDMA), or
`code division multiple access (CDMA), is a restricted resource
`for such systems, increasing demand can only be satisfied
`by diminishing the transmission power and coverage area of
`cells. This, however, increases the number of handoffs and
`the corresponding administration overhead. Hierarchical cell
`structures seem to be a good compromise between an efficient
`use of available channels while simultaneously keeping the
`number of handoffs small. Fast-moving mobiles are assigned
`to larger cells, and stationary or slow-moving stations are
`allocated to microcells, which are used in areas of high
`utilization. Of course, physical channels must be segregated
`in overlapping cell areas.
`information about
`For an efficient channel assignment,
`the position and velocity of mobile stations is inevitable in
`hierarchical networks. External data, e.g., from the global po-
`sitioning system (GPS), could be used to calculate and transmit
`the present position and velocity in subsequent time slots. This
`has two main disadvantages. First of all, the size and cost of
`mobile transmitters would be considerably increased because
`of the incorporated GPS receiver. Moreover, a clear view of
`the sky is necessary to receive usable GPS data of at least three
`satellites, which makes the system useless in buildings or roads
`surrounded by high buildings or mountains. Thus, a system is
`
`Manuscript received October 6, 1995; revised March 13, 1996. This work
`was supported by the Deutsche Forschungsgemeinschaft under Grant Ma
`1184/5-1 and Wa 542/7-1.
`M. Hellebrandt and R. Mathar are with Stochastik, Insbes, Anwendungen in
`der Informatik, Aachen University of Technology, D-52056 Aachen, Germany.
`M. Scheibenbogen is with Lehrstuhl f¨ur Kommunikationsnetze, Aachen
`University of Technology, D-52056 Aachen, Germany.
`Publisher Item Identifier S 0018-9545(97)01312-1.
`
`1997 IEEE
`
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`66
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`IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 46, NO. 1, FEBRUARY 1997
`
`for velocity estimation if diversity reception is available. The
`authors determine the speed of a mobile via the estimated
`expected diversity branch switching rate between two diversity
`branches using the Doppler effect.
`In this paper, we use a method called multidimensional
`scaling (MDS) in data analysis literature (see [1], [14], and
`[18]). According to a least squares criterion, a point in the
`area of interest is determined in such a way that the measured
`signal strength of certain base stations is best fitted to the
`known average signal strength at this point. The necessary
`data are available in the GSM system, where each 0.48 s the
`downlink signal levels of six neighboring base stations are
`transmitted on a discrete scale from 0–63. A fast algorithm
`and a corresponding smoothing procedure are developed that
`allow the online estimation of the position and velocity of
`mobiles with high accuracy. The performance of our method
`is tested by data for different complicated scenarios. These
`are generated by the powerful, close-to-reality simulation tool
`GOOSE, whose basic principles are briefly outlined in Section
`II.
`
`II. ESTIMATING POSITIONS VIA
`SIGNAL STRENGTH MEASUREMENTS
`To clarify the concept, we first introduce our method under
`simplifying assumptions. Let
`denote the position of
`base station
`and
`the Euclidean distance
`of
`from . If the average signal strength follows
`a propagation law of the type
`denoting the distance
`from the transmitter,
`a constant, and
`the attenuation
`exponent, then the average signal strength
`of base station
`measured at position
`is obtained as
`
`(1)
`
`Let
`
`denote the measured signal strength of base station
`at a certain position. Obviously,
`is subject
`to random fluctuations due to short-term Rayleigh and Rice
`fading. The transformed values
`correspond to
`the distance from base station
`such that the solution
`of
`
`minimize
`
`over
`
`(2)
`
`is a least squares estimator of the actual position at time .
`This is actually a typical question of MDS. Given certain
`pairwise pseudo distances or dissimilarities
`between
`ob-
`jects, MDS aims at determining a representation of the objects
`by
`points in a -dimensional Euclidean space such that the
`interpoint distances fit the given dissimilarities optimally. Here
`we have the special case that the positions of the base stations
`are a priori fixed, and only the position of the mobile has to
`be fitted to the transformed signal data.
`A local minimum of (2) can be calculated numerically by
`a Newton-type iteration
`
`IN
`
`(3)
`
`The gradient
`
`at differentiable points
`
`is given by
`
`(4)
`
`and with
`determined as
`
`where
`
`and
`
`the Hessian is
`
`(5)
`
`. A sufficient condition for a lo-
`Iteration (3) solves
`cal minimum at a stationary point
`is
`positive definite,
`which is checked in the following numerical examples.
`It is well known that algorithm (3) is prone to get stuck
`local minima and does not necessarily find the global
`at
`minimum. However, a reliable estimation
`of a mobile’s
`position at time
`is a reasonable starting point for finding the
`next solution
`by (3).
`The above described procedure has been applied to data
`generated by GOOSE, a tool to simulate radio wave propa-
`gation for cellular radio networks in realistic environments.
`The basic input data fed to Goose are: 1) topographical and
`morphological data; 2) mobility patterns; and 3) the cellular
`network configuration. Radio wave propagation is simulated
`on the basis of this input and follows the Okumura–Hata model
`[5], [15], taking account of reflections, shadowing, and fading.
`GOOSE is developed at ComNets at the Aachen University of
`Technology.
`The area of interest is a square of 10
`10 km that contains
`six base stations located according to Fig. 1. A plane landscape
`is assumed such that a simple propagation law with circle-
`shaped contour lines of locations of identical signal strength
`applies. A mobile is moving on a straight line from the left
`to the right margin of the area with 100 km/h, as is depicted
`by the solid line in Fig. 1. This enables measurements during
`a time period of 6 min, one each 0.48 s, in total
`blocks of
`signals strengths to each of the base stations.
`GSM mobiles code measurements of signal strength on a
`discrete scale from 0–63 (7 b). This relatively rough scale
`makes the retransformation to distances a bit more complicated
`and yields extra errors. For instance, the received coded signal
`strength from base station 2 over 360 s is depicted in Fig. 2.
`However, the test results show that our algorithm copes easily
`with these extra inaccuracies.
`Applying the Newton iteration (3) to each of the 748
`data sets,
`results in a
`sequence of successing positions that are plotted in Fig. 1.
`It can be clearly seen that estimation is more accurate in
`the middle where the mobile is surrounded by base stations.
`The starting point for the first localization by (3) is randomly
`chosen according to a uniform distribution on the relevant area.
`In each subsequent optimization, the preceding estimation is
`used as a starting point in (3). Iteration scheme (3) is stopped
`at a relative accuracy of
`.
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`HELLEBRANDT et al.: ESTIMATING POSITION AND VELOCITY OF MOBILES
`
`67
`
`If the motion of the mobile is linear with constant speed vector
`and velocity
`then the true position at time
`is given by
`
`. The parameters
`is the position at time
`where
`in this linear regression setup are estimated from the
`and
`observed values
`by the solutions
`of the
`least squares criterion
`
`minimize
`
`over
`
`(6)
`
`Minimization problem (6) decomposes into
`
`Fig. 1. Scenario, true track, and estimated positions (nonsmoothed).
`
`From standard linear regression in statistics, the solution of
`either sum is well known to be
`
`and
`with
`. Hence, a solution of (6) is given by (7) through
`
`(7)
`
`(8)
`
`Fig. 2. Received signal strength of base station 2.
`
`III. SMOOTHING ESTIMATED TRACKS AND VELOCITY
`Due to random fading of the signal, the estimated track of
`the mobile based on
`is rather zigzagged.
`This can be clearly seen from Fig. 1. The real track of a mobile
`can be approximated more closely by smoothing the data. We
`suggest a linear regression setup that has proved very efficient
`in our numerical experiences. Let
`
`denote the solution (8) based on
`let
`Now,
`the last measurements before and including time
`i.e.,
`and
`. The actual position
`is then estimated by the value of the regression line
`i.e.,
`
`at time
`at
`
`(9)
`
`simultaneously yields a smoothed
`at
`Differentiating
`estimator of the instantaneous speed vector at time
`namely
`
`denote the estimated locations at subsequent time points
`and
`IN. For the GSM system, we have
`s when
`normalizing
`. It is assumed that the motion of the mobile
`can be (at least locally) approximated by a straight line. Since
`mobiles must be localized in real time, a smoothing algorithm
`at time may only use past values at times
`for
`. We
`restrict our attention to
`preceding estimated positions and
`labeled as
`for notational convenience.
`
`The corresponding velocity is obtained as
`
`(10)
`
`(11)
`
`small with
`usually
`is
`To calculate
`the first
`
`5–20.
`between
`values
`regression coefficients
`the MDS estimators
`
`,
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`68
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`IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 46, NO. 1, FEBRUARY 1997
`
`Fig. 3. Regression-smoothed estimated track.
`
`Isoclines of signal strength of base station 2 with shadowing by hills
`Fig. 4.
`(indicated by circles).
`
`In the following, we assume that the average signal power
`is known for each base station
`and for
`each location
`of the relevant area. Let
`denote the measured signal strength of base
`station
`at a certain position. To estimate this position, a
`modified problem of type (2) arises, namely
`
`minimize
`
`over
`
`(12)
`
`estimates the
`
`A corresponding solution
`position of a mobile at time .
`To solve (12), complete knowledge of the irregular average
`power surfaces
`is necessary. The average
`signal power may be conveniently represented by a spline
`with support points
`, where
`is chosen on a
`dense grid
`in the relevant area and
`is obtained
`from a priori measurements or simulations of the average
`signal power as described above. A Newton iteration of type
`(3) yields a (local) minimum of (12), where the gradient and
`Hessian are determined according to the spline representation
`.
`is continuous and differentiable at interior points
`of the covering triangles with gradient
`
`and Hessian
`
`are missing. The necessary values are
`at time
`set equal to the first MDS-estimated position, i.e.,
`
`The linear smoothing procedure with duplicated initial es-
`timators has been applied to the above example. The corre-
`sponding estimated track is represented in Fig. 3. It shows a
`very satisfying behavior. Estimated positions could be even
`improved by additional
`information on the topography. If
`the position of roads is available from an internal map of
`the relevant area, better estimators are obtained by projecting
`onto the closest road. Such additional information on the
`environment has been used for pattern-recognition techniques
`in [9].
`
`IV. SHADOWING AND REFLECTIONS
`The above procedure presupposes an exact knowledge of
`the average signal strength following a simple propagation
`law, e.g., (1). In reality, however, this is quite rarely the case.
`Due to shadowing and reflections, the contour lines of points
`of equal average signal strength are strongly deformed in an
`environment with hills and large buildings. The isoclines are
`no longer a simple function of the distances alone, as can
`be clearly seen from Fig. 4. The scenery considered here
`is the same as above, i.e., a square of 10
`10 km with
`six base stations, but now two hills north and north-east of
`base stations 5 and 6, respectively, are inserted. The contour
`lines surrounding base station 2 are heavily influenced by this
`topography.
`the
`there exist computer programs to predict
`However,
`average signal power quite accurately and also in complicated
`environments [7], [11]. Even a three-dimensional (3-D) model
`is available in [13]. These tools may be used to determine
`the average signal strength at any position in certain sce-
`narios. Moreover, such data can also be obtained from real
`measurements while moving in the area of interest.
`
`and
`
`by
`
`yields (4) and
`
`by
`Substituting
`(5) as a special case.
`This type of fitting the position via a spline function worked
`well in numerical examples, but it is relatively calculation
`insensitive and time consuming. For practical purposes, it
`is sufficient to solve (12) over
`on a grid
`with
`grid constant 25 m, say, in both coordinates. Then, (12) is a
`finite optimization problem. After having determined a reliable
`
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`HELLEBRANDT et al.: ESTIMATING POSITION AND VELOCITY OF MOBILES
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`69
`
`Fig. 5. Regression-smoothed estimated track with shadowing.
`
`Fig. 6. Regression-smoothed estimated velocity with shadowing.
`
`subsequent minima must be searched only
`estimator
`over a local square of grid values in
`surrounding
`since the mobile’s distance from
`is bounded by the
`the maximum velocity, and
`the time
`radius
`between subsequent measurements. The minimization can be
`carried out on-line by complete enumeration and yields a
`fast method to trace the mobile by estimates
`. Position and velocity are then determined by
`the smoothed regression estimators with (9) and (10) with
`. Fig. 5 shows the resulting track for the environment
`of Fig. 4. An average deviation of 95 m from the true track is
`observed that is quite satisfying for practical purposes. Fig. 6
`shows the estimated velocity (in meters per second) over time
`(from 0 to 360 s), based on the regression-smoothed values
`(11). Although averaging over the last 15 measurements, the
`estimation error is still quite large. Large deviations from
`the constant speed of 27.8 m/s occur, demanding further
`smoothing.
`
`V. TEST RESULTS FOR NONSTRAIGHT
`MOTION AND FURTHER SMOOTHING
`The above described procedure, based on a linear regression
`setup,
`is adapted to at
`least
`locally straight motion. It
`is
`important to see how bended tracks influence its performance.
`For this purpose we consider a relevant area of 10
`10
`km with seven base stations, whose positions are depicted in
`Fig. 7. Moreover, the average signal-strength isoclines of base
`station 2 are included, corresponding to two hills located at
`coordinates (0, 0) and (4.8, 4.8). A mobile is moving from the
`left to the right margin with a constant speed of 100 km/h on
`a track containing a quarter right, two-quarter left, and again a
`
`Fig. 7. Base stations, the mobile’s track, and isoclines of base station 2.
`
`quarter right turn. The mobile can be observed for 9 min in the
`relevant area. Its way is represented by a solid line in Fig. 7.
`have been esti-
`Successive positions
`mated by solving the discrete optimization problem described
`in Section IV. Based on these values, the regression-smoothed
`from (9) are calculated and successively
`estimators
`connected by lines. The corresponding estimated track of the
`mobile is depicted in Fig. 8. The average deviation from the
`true track is 90 m. In spite of three bends and constant speed
`while turning, the estimators are quite accurate and reliable.
`By now we have not used the fact that a mobile of maximum
`cannot leave the circle of radius
`speed
`surrounding its present position within time . This information
`
`if
`otherwise.
`
`(13)
`
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`70
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`IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 46, NO. 1, FEBRUARY 1997
`
`Fig. 8. Regression-smoothed estimated track for the scenario of Fig. 7.
`
`Fig. 10. Projection-smoothed estimated velocity.
`
`using (11). Estimated values are still sensitive against random
`fluctuations of signal strength, as can be seen from Fig. 10.
`Compared to Fig. 6, however, a much improved behavior
`is achieved. In the middle area, where the mobile is fully
`surrounded by base stations, the maximum deviation from the
`true velocity is about 10 m/s.
`
`VI. CONCLUSIONS
`A method, resembling MDS, has been proposed to estimate
`the position and velocity of mobile stations from signal-
`strength measurements of the downlink to reachable base
`stations. Two smoothing procedures are employed to gain
`accurate values, based on a linear regression setup and a
`projection method onto the ball of maximum deviation from
`the actual position. Tests in a complicated scenario show that
`an average deviation of 60 m in location and a maximum
`deviation of 10 m/s in velocity can be achieved. This makes the
`method applicable as a decision support for the assignment of
`channels in hierarchical networks with macro and microcells.
`Estimation can also be based on radio wave propagation
`time from base stations to mobile transmitters. This approach
`will be investigated in future work.
`
`ACKNOWLEDGMENT
`The authors appreciate the valuable suggestions of two
`anonymous contributors.
`
`REFERENCES
`
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`Fig. 9. Projection-smoothed estimated track.
`
`of
`and
`is used to develop refined estimators
`position and velocity, respectively, which are described in the
`following.
`denote the refined position estimator at time
`Let
`is determined by solving (12), using
`.
`is
`as an initial value, as described in Section IV.
`and center
`then projected onto the ball with radius
`yielding (13), as shown at the bottom of the previous
`is estimated
`page. The position of the mobile at time
`values
`by the regression approach (6)–(9) using the last
`.
`This approach, named projection estimation, has turned
`out to work extremely well in the scenario of Fig. 7. The
`maximum speed is set to 250 km/h, corresponding to
`m with
`s. The estimated projected track
`is depicted in Fig. 9. There is only a small
`tendency to
`overswing when the mobile makes a quarter turn. The average
`deviation from the true track is 62 m, which is fully acceptable.
`Corresponding velocities have been treated the same way
`
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`HELLEBRANDT et al.: ESTIMATING POSITION AND VELOCITY OF MOBILES
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`71
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`
`Martin Hellebrandt was born in Geldern, Ger-
`many, in 1969. He received the Dipl. degree in
`mathematics from the Aachen University of Tech-
`nology, Aachen, Germany, in 1994.
`He is currently working at the Department of
`Stochastics at the Aachen University of Technology.
`His research interests include mobile communica-
`tion systems, queueing theory, applied probability,
`and optimization.
`
`Rudolf Mathar was born in Germany in 1952.
`He received the Dipl.Math. and Dr. Rer.Nat. de-
`gree in mathematics from the Aachen University of
`Technology, Aachen, Germany, in 1978 and 1981,
`respectively.
`From 1986 to 1987, he worked at the European
`Business School as a Lecturer in Computer Science,
`and in 1988 he joined an applied optimization
`research group at the University of Augsburg. In
`October 1989, he joined the faculty at the Aachen
`University of Technology, where he is currently a
`Professor of Stochastics. He is especially interested in applications to com-
`puter science. His research interests include mobile communication systems,
`performance analysis and optimization of networks, and applied probability.
`He is the author of over 50 research publications in the above areas.
`
`Markus Scheibenbogen received the Diploma degree in electrical engineering
`in 1994 from the Aachen University of Technology, Aachen, Germany. He is
`currently working towards the Ph.D. degree in dynamic channel assignment.
`In 1994, he joined the Department of Communication Networks (COM-
`NETS), Aachen University of Technology, where he works in a research
`group for a wireless ATM air interface and the implementation of protocols in
`a wireless ATM demonstrator. His research interests are in the fields of mobile
`communication protocols, dynamic channel allocation, especially for wireless
`ATM systems, and the determination of guardbands for mobile communication
`systems.
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