A Polygon to Ellipse Transformation Enabling Fingerprinting and Emergency Localization in GSM

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A Polygon to Ellipse Transformation EnablingFingerprinting and Emergency Localization in GSM

Torbjörn Wigren, Senior Member, IEEE

Abstract—Cellular location-based services need to use a variety ofpositioning methods to meet requirements on availability, response time,and accuracy. Due to the geometrical properties of the different positioningmethods, the number of standardized reporting formats is large. Becausethe user requirements on the reporting formats are as scattered as thegeometrical properties of the positioning methods, there is a need for shapeconversion algorithms that transform between different formats. This pa-per discusses a case that transforms between a polygon and an ellipse whilepreserving the confidence between the two shapes. The transformationis required for emergency positioning outside North America, and whenfingerprinting, localization is used in the Global System for Mobile Com-munications (GSM). The algorithm has a number of technically interestingproperties, e.g., with regard to the selection of the optimization criterionand the fact that the solution is computed by an analytical solution of acubic equation.

Index Terms—Cell identity (cell ID), confidence, cubic equation,E-911, E-112, ellipse, fingerprinting, geometry, Global System for MobileCommunications (GSM), localization, navigation, polygon, positioning,signaling.

I. INTRODUCTION

The dominating cellular localization technology that is deployedfor E-911 emergency positioning in the U.S. is the assisted GlobalPositioning System (A-GPS) [1]–[5], which is an enhancement of

Manuscript received August 5, 2010; revised October 18, 2010 andJanuary 7, 2011; accepted February 21, 2011. Date of publication February 28,2011; date of current version May 16, 2011. The review of this paper wascoordinated by Dr. H. Lin.

The author is with Ericsson AB, WCDMA System Management, 16480Stockholm, Sweden (e-mail: torbjorn.wigren@ericsson.com).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2011.2120637

the U.S. military global positioning satellite navigation system; forexample, see [6]. An A-GPS system exploits GPS reference receiversthat track the GPS ranging signals from each GPS space vehicle (SV).The retrieved information is relayed to the GPS-capable terminals asassistance data. The terminals can then skip decoding steps, whichspeeds up SV signal acquisition and enhances receiver sensitivity [7].However, even with the enhancements provided by assistance data andfine time assistance (TA) [8], A-GPS has a limited indoor coverage.Therefore, cellular systems also implement alternative localizationtechnologies. Terrestrial time difference of arrival (TDOA) positioningmethods have, for example, extensively been studied [3], [4], [9]. Anuplink version of TDOA is, for example, well established for E-911positioning in the Global System for Mobile Communications (GSM)[5]. Low- and medium-accuracy positioning methods such as cellidentity (cell ID), cell ID and timing advance (TA) in GSM, and roundtrip time (RTT) in wideband code-division multiple access (WCDMA)are more common. In WCDMA, cell ID positioning determines theposition of the terminal with cell granularity, reporting a prestoredgeographical cell description from the radio network controller (RNC)to the core network (CN) [10]. In RTT positioning, the travel time fromthe radio base station (RBS) to the terminal and back is measured andused for the computation of the distance between the known positionof the RBS and the terminal [11]–[15]. It is also possible to combinecell ID and RTT information, as shown in [13]–[15]. Another recenttechnology is the fingerprinting class of methods; for example, see[16]–[18]. These methods use radio signatures that were measured bythe terminal. The radio signatures are compared with the signaturesand locations of a database, reporting the location that best fits themeasured radio signature.

A-GPS positioning generates points in WGS84 coordinates, to-gether with ellipse or ellipsoidal uncertainty measures [19]. Cell IDand adaptive enhanced cell identity (AECID) fingerprinting [16], [17]require the flexible polygon format [19] to describe complicated re-gions. At the receiving end, diverging format requirements complicatethe reporting of the obtained geographical shape to the CN. In the U.S.,it is, for example, regulated that the emergency centers will receivepositions as points with an uncertainty circle, although this conditioncreates a severe loss of accuracy [20]. In European countries, the sameapplication requires the reporting of a point with uncertainty ellipse.There is, hence, a need for transformations between reporting formats.This functionality is denoted shape conversion.

The main contribution of this paper is an algorithm that transformsa polygon to a point with uncertainty ellipse. The first motivation forthis paper is that GSM lacks full support for the reporting of polygons,which would prevent the use of the algorithm in [16] and [17] in GSM.The second motivation is the requirement of using uncertainty ellipseinformation in emergency positioning in some European countries.Because the ellipsoid arc shape associated with RTT positioning canoften be well approximated by a polygon, algorithms such as theproposed approach are hence required to use cell ID, RTT [14], [15],and fingerprinting [16], [17] in these countries. One important benefitof the proposed method is that it preserves the confidence between theformats. The major and minor axes of the ellipse are obtained froman analytical solution to a cubic equation, with the cubic equationresulting from a tailored criterion function, which avoids problemswith multiple branches of the ellipse.

Previous work on an optimal way of inscribing a polygon in anellipse was already performed in 1944 [21]. Such algorithms, however,do not solve the problem at hand. The singular-value decomposition ofthe covariance of the corner points cannot stringently handle confi-dence. Finally, note that the analytical solution of the cubic equation

0018-9545/$26.00 © 2011 IEEE

1972 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 4, MAY 2011

used here was independently obtained in Italy in the 16th century [22],[23] by S. del Ferro and N. Tartaglia.

This paper is organized as follows. Section II treats reportingformats. The transformation is derived in Section III. A numericalexample is given in Section IV, and the conclusions in Section V endsthe paper.

II. REPORTING FORMATS

A. Confidence Model

Due to the nature of radio propagation, it is standard to adopt astatistical description of the obtained positions of the terminals. Theconfidence, i.e., the probability that the terminal is located in theinterior of the reported region, is used to characterize the positioningerror. The ways that the confidence is obtained differ due to differentstatistical models for different positioning methods. In A-GPS, theinaccuracy is caused by pseudorange measurement errors [6] andgeometrical effects. The law of large numbers, together with a lin-earization, then gives a Gaussian position error model. For cell ID, cellID+TA, RTT, and fingerprinting, the error is rather caused by the casethat the position of the user is determined to be somewhere within therelatively large region of the cell or the fingerprinted region. Becausethere is normally no information with regard to the distribution ofusers within these regions, it is natural to adopt a uniform statisticalmodel (flat prior) for the terminal location. This case motivates whythe distribution associated with the ellipse is treated uniform in thispaper.

B. Polygon and Ellipse Formats

The polygon is described by a list of 3–15 latitude, longitudecorners, encoded in WGS 84 coordinates; see [19] and Fig. 4 fordetails. The polygon format does not carry confidence information[19], and a configurable parameter can be used for this purpose. TheThird-Generation Partnership Project (3GPP) point with the uncer-tainty ellipse format (see Fig. 4) is parameterized with semimajor axisa, semiminor axis b, and an angle ϕ relative to the north, countedclockwise from the semimajor axis. The format carries confidenceinformation [19].

III. TRANSFORMATION

The task is to transform the polygon format in [19], together with aconfidence value, to the ellipsoid point with uncertainty ellipse formatin [19], together with a different confidence level. The associatedprobability distributions are assumed uniform for the reasons outlinedin Section II. In the description that follows, the corners of the polygonare assumed to have been decoded from the format in [19] andtransformed to a local earth-tangential Cartesian coordinate system,with coordinate axes in the east and north directions; see [14]. Theellipse to which the polygon will be transformed is described by the3GPP format. In the description that follows, the angle ϕ, which isdefined with respect to the north, is replaced by α = π/2 − ϕ to workin a right-hand coordinate system.

A. Processing of the Polygon

1) Computation of the Area of the Polygon: The area is computedby integration between adjacent corners of the polygon, with thesecorners expressed in a right-hand local earth tangential coordinatesystem. The corners are collected in

rp =

(x1 . . . xNp

y1 . . . yNp

). (1)

Here, Np denotes the number of polygon corners. The polygon areaAp is then obtained as [24]

Ap =1

2

Np−1∑i=1

(xiyi+1 − xi+1yi) +1

2

(xNpy1 − x1yNp

). (2)

2) Computation of the Center of Gravity of the Polygon: The centerof gravity is denoted by (xCG yCG)T . Standard results, again basedon the integration between adjacent polygon corners, give [24]

xCG =1

6Ap

(Np−1∑

i=1

(xi + xi+1)(xiyi+1 − xi+1yi)

+ (xNp + x1)(xNpy1 − x1yNp)

)(3)

yCG =1

6Ap

(Np−1∑

i=1

(yi + yi+1)(xiyi+1 − xi+1yi)

+ (yNp + y1)(xNpy1 − x1yNp)

). (4)

3) Search for the Orientation of the Polygon: To avoid a com-putationally intense joint numerical search over all parameters ofthe ellipse, a scalar search for the orientation α of the polygon isperformed. The idea is to search over a set of lines that pass throughthe center of gravity of the polygon to find the line with the longest linesegment with end points on the boundary of the polygon. This searchis performed by the following algorithm.

1) Select test angles (uniformly in [−π/2, π/2]), defining theslopes of lines that pass through the center of gravity of thepolygon.

2) For each of the lines (angles) that pass through the center ofgravity of the polygon:a) Determine all intersections between the line through the

center of gravity and the line segments that form the polygonboundary.

b) Determine the longest line segment, which is defined bythe line that passes through the center of gravity and theintersections.

3) Select the angle α as the angle that generates the longest linesegment.

To mathematically formulate this approach, ri and rj are used todenote two adjacent corners of the polygon. It then follows based onFig. 1 that the point r on the boundary of the polygon fulfills (5) and(6), where

r = rCG + β

(cos(α)sin(α)

)(5)

r = ri + δ (rj − ri) (6)

and β and δ are scalar parameters. The solution to the system ofequations (5), (6) is given as follows:(

βδ

)=

(cos(α) rx

i − rxj

sin(α) ryi − ry

j

)−1(rx

i − xCG

ryi − yCG

). (7)

The superscripts x and y denote the x- and y-components of avector, respectively. For a given α and pair of corner points of thepolygon (ri and rj), the parameters β and δ are determined. In thecase that δ ∈ [0, 1], the intersection falls between the corner pointsand is valid; otherwise, it is discarded. The calculation of β and δ are

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Fig. 1. Geometry used to determine α.

repeated for all line segments of the boundary. Because the directionvector of the line through the center of gravity is normalized, it followsthat the length of the line segment between the center of gravity andthe boundary is given by the corresponding β. The intersections k(maximum length) and l (minimum length, other direction), whichgenerate the largest difference, i.e.,

l(α) = βk − βl (8)

correspond to the sought candidate length for the angle α. Finally, theangle α that renders the largest value of l(α), for all α, is determined,i.e.,

αe = arg maxα

l(α) = π/2 − ϕ. (9)

4) Translation and Rotation of the Polygon: As a preparation forthe subsequent optimization step, the polygon corners are translatedso that the center of gravity of the polygon is moved to the origin.They are then rotated so that the orientation of the determined polygonaxis, with angle αe, coincides with the x-axis. The result is

r′p = rp − rCG (10)

r′′p =

(cos (αe) sin (αe)− sin (αe) cos (αe)

)r′p. (11)

Here, r′p and r′′p denote the translated and rotated polygon coordi-nates, respectively.

B. Optimization of the Ellipse

To describe the calculation of the semimajor and semiminor axes,the confidence of the polygon is denoted as Cp. The desired confidencevalue of the ellipse that will be computed is denoted by Ce. Asmotivated in Section II-A, the distribution of users is assumed uniformover the polygon and the ellipse; hence, the following constraint holdsfor the areas of the polygon Ap and the ellipse Ae:

Ae =Ce

Cp

Ap. (12)

Then, because the area of an ellipse is πab, where a and b denotethe semimajor and semiminor axes, it holds that

ab =1

π

Ce

CpAp. (13)

Equation (13) provides a constraint that can be used to eliminate ei-ther a or b from the following optimization problem, thereby reducingthis problem to a 1-D case.

The algorithm now calculates the semiminor axis that provides thebest fit according to the following criterion:

V (b) =1

Np

Np∑i=1

((y′′

i )2 − (ye (b, x′′

i ))2)2

(14)

where the ellipse model (ye(b, x′′i ))2 follows from

x2

a2+

y2

b2= 1. (15)

Remark: Note that the square of the y-coordinates of the polygonand the ellipse model are compared in (14). If the y-coordinateswould not be squared, both the positive and negative branch of (15)(y = ±

√b2 − b2x2/a2) would need to be treated in a minimization,

together with square-root handling. The additional logic would havecomplicated the algorithm. Considering optimality, a straightforwardcriterion that is not based on differences between squared quantitieswould result in another solution. It is an open question, which is thebest choice; however, the use of squared quantities makes (14) moresensitive to large errors in single polygon corners. This case willlikely produce solutions with a more uniform distribution of errorsbetween polygon corners, rendering a solution that is closer to whatwould result from an ‖ ‖∞ optimization. Robust performance is henceanother motivation for (14).

When (15) is backsubstituted in the criterion (14), the optimizationproblem can be posed as

b=

√√√√arg minb2

1

Np

Np∑i=1

((y′′

i )2−(b2)+(b2)2π2C2

p

C2eA2

p

(x′′i )2)2

(16)

after the elimination of a using (13). A differentiation of the sum ofsquares, with respect to b2, renders the following cubic equation forb2, from which b2 can be solved:

ε0 + ε1b2 + ε2

(b2)2

+ ε3

(b2)3

= 0 (17)

ε0 = −Np∑i=1

(y′′i )

2 (18)

ε1 =

Np∑i=1

(1 + 2

π2C2p

C2eA2

p

(x′′i )

2(y′′

i )2

)(19)

ε2 = −Np∑i=1

3π2C2

p

C2eA2

p

(x′′i )

2 (20)

ε3 =

Np∑i=1

2π4C4

p

C4eA4

p

(x′′i )

4. (21)

To optimize the numerical results, it is recommended that (12)–(21)are scaled with the polygon radius.

The cubic equation is analytically solved using the techniques ofHarriot [22], [25]. Using u = b2, and a division of (17) with ε3,gives

u3 + eu2 + fu + g = 0. (22)

1974 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 4, MAY 2011

Fig. 2. General positioning mode of the AECID method in GSM. Received signal strength indicators (RSSIs) are not used in the simulations of Example 1.

Equation (22) is then transformed to a depressed cubic, without asecond-degree term [22], by

u = t − e

3. (23)

The transformations

p = f − e2

3, q = g +

2e3 − 9ef

27, t = z − p

3z(24)

then results in the following quadratic equation in z3:

(z3)2 + qz3 − p3

27= 0. (25)

The solutions of the original cubic problem now start with a com-putation of the coefficients of (18)–(21), followed by a computationof e, f , and g of (22). Based on this information, p and q of (24) areobtained. Equation (25) is then solved, and the two solutions for z3

are recorded, after which, all six candidate solutions for z are obtainedby a complex third-root extraction. The backsubstitution of all rootsz → t → u → b2 results in six candidate solutions for b2 = u. Threecorrect solutions are obtained by backsubstitution and checking for thefulfillment of (17). In case only one solution is real (and positive), b iscomputed from the corresponding square root. In case more than onereal and positive solution is found, the solution that renders the leastvalue of (14) is selected.

With b determined from (14)–(25), the final step of the algorithmgives a based on (13) as

a =1

π

Ce

CpAp

1

b. (26)

An extension, which improves the numerical results, modifies theconfidence according to

CScaledp = SCp (27)

where S is a scale factor. Without (27), the ellipse area may sometimesbe placed along the semimajor axis, outside the polygon, where thereis no penalty of the criterion function that counteracts this tendency.After optimization, the semimajor and semiminor axes are rescaledaccording to

aRescaled =√

Sa, bRescaled =√

Sb. (28)

IV. NUMERICAL ILLUSTRATION

A detailed description of the AECID method appears in [16] and[17]. Here, a brief summary is given, assuming that cell IDs and TAs,

Fig. 3. Color-coded fingerprinted clusters. Different colors correspond todifferent fingerprints. Each dot represents an A-GPS position. Base stationpositions are marked “∗,” and ideal cell boundaries are represented as solidlines.

but not signal strengths, will be used for fingerprinting. WheneverA-GPS is performed in the cellular system, AECID collects theresulting A-GPS position. At the same time, AECID registers the cellID and measures the TA, thereby providing a fingerprinted A-GPSposition. The algorithm then collects all A-GPS positions with thesame fingerprint in separate clusters. AECID proceeds by calculatingthe cluster boundary in terms of a 3GPP polygon for each cluster[16], [17]. The result of this database buildup mode of AECID isa database of fingerprinted polygons. In the positioning mode ofAECID (see Fig. 2), only a fingerprint is determined, after which, thefingerprinted polygon is found in the database and is reported. In GSM,the shape conversion of this paper is also used.

Example: Here, the accuracy of AECID is not discussed; see [26]for field trial results. The performance of the shape conversion ratherconcerns the evaluation of a mathematical approximation algorithm.Hence, simulated data, at the AECID polygon stage, are believed to besufficient. The complete Ericsson MATLAB reference implementationof the AECID fingerprinting positioning chain was used to generatethis result, exploiting A-GPS for reference position collection, and cellID and TA for fingerprinting, as described in [26]. A nonhexagonalsix-GSM-base-station scenario was used to generate the fingerprintedclusters of A-GPS positions, which is shown color coded in Fig. 3. TheA-GPS positions were nonuniformly distributed with higher densities

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 4, MAY 2011 1975

Fig. 4. Examples of the results of the shape conversion algorithm. Thepolygon is solid black, the ellipsoid point is marked with a “∗,” and the ellipseis solid blue.

Fig. 5. Distribution of the number of polygon corners over the 40 polygons ofthe simulated example.

Fig. 6. CDF of the shape conversion error for each polygon corner (green)and of the maximum shape conversion error over all corners of each polygon(blue). The errors are normalized with the semimajor axis of each computed3GPP ellipse.

along freeways and in hot spots. Each A-GPS position was associatedwith an ideal cell after the addition of shadow fading of about 5 dB.Overlapping TA regions provided each A-GPS position with the TApart of the fingerprint.

After clustering, fingerprinted 3GPP polygons were computed. Theshape conversion was then applied. Fig. 4 depicts two typical examplesof the shape conversion result. Here, the algorithm used the parametersCe = 0.95, Cp = 0.8 (a), Cp = 0.70 (b), and S = 1.0. The ellipsescomputed quite accurately cover the polygons. This result is generallythe case, as illustrated in Figs. 5 and 6, which show the number ofpolygon corners and the cumulative distribution function (cdf) of theshape conversion error, over all 40 polygons and ellipses. The erroris expressed as the minimum distance between each polygon cornerand the corresponding ellipse. The cdf of the maximum error over allpolygon corners, for each polygon, is also given in Fig. 6. The fieldtrial of reference [26] further supports these conclusions.

V. CONCLUSION

This paper has proposed an algorithm for transformation from the3GPP polygon to the 3GPP ellipsoid point with uncertainty ellipse.The scheme is a requirement for the use of AECID fingerprinting inGSM and of cell ID, RTT positioning, and AECID fingerprinting forE-112 services in some European countries. As shown in this paper,the parameters are obtained from the solution of a cubic equation. Thepossibility of analytically solving this equation is a further advantagewhen the selection is made among possible multiple solutions. Theunderlying criterion is expected to provide better robustness thanrelated algorithms. The algorithm is a part of a commercial position-ing system, which is deployed at a major North American cellularoperator.

ACKNOWLEDGMENT

The author would like to thank A. Kangas of Ericsson Research fortechnical discussions, F. Zhao, Q. S. Cheng, and other staff of Ericsson,Shanghai, China, for the skillful porting from the MATLAB referenceimplementation to C++, which greatly contributed to the final result,and M.Sc. E. Schön for the permission to publish this paper.

1976 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 4, MAY 2011

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