2010 IEEE 4th International Symposium on Advanced Networks and Telecommunication Systems

Adaptive Distance Estimation and Localization in Wireless

Networks with Triangle and Ptolemy Inequalities

Pejman Khadivi

Department of Electrical and Computer Engineering, Isfahan University of Technology, 84156-83111, Isfahan, Iran. E-mail: pkhadivi@cc.iut.ac.ir

Abstract- Distance estimation and localization are essential tasks

in modern wireless networks with a wide range of applications.

Distance is a geometrical concept and hence, various numbers of

geometric inequalities and theorems can be applied to distance

measurement. In this paper, new adaptive approaches for

distance estimation and localization problems are proposed based

on Triangle and Ptolemy's inequalities. Simulation results are

presented which show that the proposed methods can improve the

performance of the existing approaches, in terms of accuracy.

I. INTRODUCTION

Distance estimation is considered as one of the important tasks in wireless communications. This task is the basic functionality in localization which should be considered indispensable these days. Location-aware services and emergency calls are two important domains where localization is applicable [1]. Beside localization, distance estimation is also applicable in routing algorithms [2], [3].

Different solutions have been proposed for distance estimation in the literature [I], [4]. Some well-known solutions are distance estimation based on received signal strength (RSS) and time-of-arrival measurements. Also, various centralized and distributed algorithms have been proposed for localization in wireless networks [5], [6]. There are also solutions for this problem in different wireless environments, including underwater sensor networks [7] and millimeter wave communications [8]. In cellular networks, traditionally, localization can be performed based on the exact location of Base Stations (BS). This approach is named as localization based on distance-ta-references. This method of localization is also applicable in wireless LANs (WLAN), where BSs are replaced by access points (AP) [1]. Another method of localization in wireless environments is to employ global positioning system (GPS) devices. These days, a large number of wireless nodes are equipped with GPS receivers and hence, they can localize themselves. Known as anchors, these nodes can be employed in localization of other wireless devices [5].

Extensive works have been reported in the literature considering the accuracy of localization based on RSS measurement [9], [10]. Important sources of distance estimation error are measurement noise, and Non-line of sight (NLOS) communication. However, some considerations about the geometry of the wireless network can be helpful to construct more accurate methods [1].

In this paper, new distance estimation and localization algorithms are proposed based on two well-known inequalities of Euclidean geometry. The proposed adaptive methods, with

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the help of Triangular and ptolemy's inequalities, improve the behavior of the estimators in terms of accuracy. Simulation results show that by employing these inequalities, estimation error can be reduced down to an acceptable level.

The remainder of the paper is organized as follows. Section IT highlights the importance of geometric inequalities in distance estimation. New distance estimation algorithms are proposed in Section TTT. In Section IV, a new localization approach is proposed. Simulation results are presented in Section V. Section VI is dedicated to some concluding remarks.

II. DISTANCE ESTIMATION AND GEOMETRIC INEQUALITIES

Principally, distance is a geometric quantity. In Euclidean geometry, there are theorems and inequalities that place constraints on this concept. Examples include Triangular Inequality (TT) and Ptolemy'S Inequality (PI) which are correct for triangles and quadrilaterals, respectively. Triangle inequality holds for any triangle and states that the sum of the lengths of any two sides must be greater than the length of the third one. In other words, for the triangle of Fig. I (a), we have:

(I)

ptolemy'S inequality is an extension of ptolemy'S theorem for any given quadrilateral. Based on PI, for a given quadrilateral such as ABCD, illustrated in Fig. 1 (b), the following inequality holds:

(2)

While TI and PI are general rules about distance in triangles and quadrilaterals, they can be helpful in distance estimation. It should be noted that not every three random numbers can be considered as the side-lengths of a triangle. This is also true for any six random numbers which are supposed to be the lengths of sides and diagonals of a quadrilateral. Experiments show that if someone picks up six random numbers, uniformly distributed over [0,100] , PI holds

with the probability of 0.748. As an example, assume that in Fig. I(a), dAc = 2 and dBC = 5 . Then, based on TT, dAB should

be less than 7 and greater than 3. Now, let us assume that we want to estimate d = dAR and the exact length of dAR is 5. No

matter which distance estimation technique is in use, with the help of TI, the estimation error can be limited to 2. In the following section, two new algorithms are proposed based on TI and PI for distance estimation in wireless networks.

d � dAIl 8

� (a) (b)

Fig. I (a) Example triangle, and (b) quadrilateral.

m. DISTANCE ESTIMATION ALGORITHMS

In this section, approaches are proposed for distance estimation with the help of geometric inequalities Tl and PI. For simplicity, the proposed approaches are based on RSS measurement. It should be noted that the same idea can be used with other methods of distance estimation.

In this paper, a log-normal path loss propagation model with shadow fading is assumed for wireless channels which is consistent with the literature [11], [12]. Each RSS sample is expressed as

RSS[k] = PT - L -lOa log(d[k]) + F(J"[k] (3)

where, k is the time index, Pr is the transmitted power, L is a constant power loss, a is the path loss exponent, and F(J"[k]

represents the shadow fading, modeled as a zero mean Gaussian random variable with standard deviation Ci. The distance between the transmitter and the receiver is d[k] . With

the traditional RSS based distance estimation model, N

consecutive samples of RSS are measured and averaged to overcome the effect of shadow fading. Then, d is estimated by solving the following equation

RSSavg = PT - L -lOa log(d) (4)

where, RSSAvg is the mean RSS averaged over N samples. It is almost obvious that estimation accuracy depends on the standard deviation of shadow fading in the wireless channel.

A. Distance Estimation with Triangle Inequality Let us assume that we want to estimate the distance

between wireless nodes A and B, and a third wireless node, such as C, is available. Nodes A, B, and C, construct a triangle. Then, Tl must hold for sides of triangle MBC. This situation is illustrated in Fig. I (a). The proposed error reduction strategy can be described as follows:

1. Each node estimates its distance with the other two nodes. Temporary estimates are illustrated by dAB, dAc, and

d'sc for AB, AC, and BC sides, respectively.

2. The following three inequalities are evaluated:

dAB < dAc + d'sc dAC < dAB + d'sc d'sC < dAB + dAc

(5)

3. If all the inequalities of (5) hold, dAB is considered

as the final estimate of d. Otherwise, steps I and 2 are repeated.

89

ALGORITHM TI Based Distance Estimation - - -

INPUT: Wireless Nodes A, B, C; OUTPUT: Estimated Distance Between A and B; BEGIN

01 WHILE TRUE DO

02 NewRSS_AB � N RSS samples at A from B;

03 NewRSS_AC � N RSS samples at A from C;

04 NewRSS_BC � N RSS samples at B from C;

05 RSS_AB � RSS_AB U NewRSS_AB;

06 RSS_AC � RSS_AC U NewRSS_AC;

07 RSS_BC � RSS_BC U NewRSS_BC;

08 AvgRSS_AB � mean (RSS_AB) ;

09 AvgRSS_AC � mean(RSS_AC);

10 AvgRSS_BC � mean(RSS_BC);

11 Temp_AB � Distance (AvgRSS_AB);

12 Temp_AC � Distance (AvgRSS_AC);

13 Temp_BC � Distance (AvgRSS_BC); 14 IF Triangular Inequalities hold THEN

15 RETURN Temp_AB;

16 END IF

17 IF Number of Samples > Sampleupper Bound THEN

18 RETURN Temp_AB; 19 END IF

20 END WHILE

END

Fig. 2 Proposed distance estimation method based on Tl.

While the introduced solution can improve the distance estimation accuracy, a large number of RSS samples may be measured by each participating node. On the other hand, by increasing the value of N in the basic RSS-based approach, one can increase the accuracy of estimation. The pseudo-code of an improved version of the proposed distance estimation method is illustrated in Fig. 2. In this algorithm, instead of considering only the last N samples, all the measured samples of RSS are considered in the estimation process. However, since this process may take a long time, an upper bound is applied on the number of samples. This upper bound is named as SampleupperBound' Distance function in lines 11-13, estimates the distance based on Eq. 4.

B. Distance Estimation with Ptolemy's Inequality In the last subsection, distance estimation was performed

with the help of TT. A similar approach can be constructed based on PI. Assume that we want to estimate the distance between the wireless nodes A and C, in Fig. I (b), while wireless nodes, B and D are available. Then, PI must hold for sides and diagonals of quadrilateral ABCD. In other words

dAB x dCD + dAD x d BC 2 d x d BD (6)

where, d is the distance that must be measured. If this inequality does not hold, it means that the measured distances have unacceptable errors and distance estimation must be repeated until (6) holds.

IV. LOCALIZATION WITH TRIANGLE INEQUALITY

In this section, a new approach for localization is proposed based on TI. The basic approach of localization is distance-to­references method. Fig. 3(a) illustrates this traditional method. In this figure, we want to fmd the location of wireless node M.

It is assumed that three anchor nodes, A, B, and C, are available. Then, distance between M and each of the anchor nodes is measured. Node M is on the intersection of three circles centered at the anchor nodes.

B

B

(a) (b)

Fig. 3 (a) Traditional localization with distance to references method. (b) Proposed solution for localization.

� 0.25

.......... :,.. .....

� 0.2 ....

.... ,..0.-;.; .... � ....... .. 0:: 0.15 ;,._ .. ;.:-,; .... 0'

.. .","", ..... 0-

0.1 ;6''' .. -

5 10 15 Shadow fading standard deviation

Fig. 4 Relative error of distance estimation for proposed method with TI and PI compared with the traditional approach.

1- - -Simple Estimate l 70 - Estimate with TI I 6 0

� 50

� 40

E � 30

.. ';;

,

....

10�---�

°O��,--72�3��4--5��6--77�8�7-�· Shadow fading standard deviation

Fig. 5 Localization error of the proposed method with TI compared with the traditional approach.

Let us assume that in addition to the anchors, an extra wireless node, N, is also available. This situation is illustrated in Fig. 3(b). As it is illustrated in this figure, different triangles could be constructed, where triangles f..MAN, f..MBN, and f..MCN are employed in the proposed solution. Similar to the distance estimation method, proposed in Section TTT.A, estimated distances must obey the Tl rule. Hence, for each of the mentioned triangles, three TT rules should be evaluated. Estimated location is appropriate if the measured distances satisfY the TT rules. Otherwise, the whole process is repeated until Tl rules are satisfied or an upper bound of samples reached. Simulation results presented in Section V show dramatic improvements in the accuracy of the localization.

V. SIMULATION RESULTS

A large variety of simulations have been performed with a wide range of parameter settings. In this section a number of most representative results are presented which illustrate the relative performance of the proposed approaches.

In the simulations, path loss propagation model of Eq. 3 has been considered. It is assumed that a is 3.7, L is 28.7 dB, Pl is 100 m W, and (J varies on [0,15] dB. Upper bound of the

90

number of samples is 15000, and N is 5. For distance estimation, four nodes are uniformly distributed over a 100 xl 00 meters area and experiments are repeated 25000 times. For localization, three reference points are located at (0,0), (200,200), and (200,0). Two nodes are unifonnly distributed over a 20x60 meters area, centered at (50,100). Simulation results are presented in Figures 4 and 5.

Relative errors of distance estimation with the proposed method based on Tl and PI are illustrated in Fig. 4. It is almost clear that the proposed methods can improve the estimation error, comparing with the traditional method. Fig. 5 shows the localization error versus (J for the traditional and the proposed methods. It is clear that the proposed solution can dramatically decreased the estimation error.

VI. CONCLUSIONS

In this paper, we proposed new approaches for distance estimation and localization problems. The proposed methods are based on the Triangle and Ptolemy's inequalities. Simulation results show that the proposed approaches improve the behavior of the estimation process in terms of accuracy. However, it can be seen that distance estimation with Triangle Inequality results in more accurate results than the approach based on Ptolemy's Inequality.

REFERENCES

[1] A.H.Sayed et al. "Network-Based Wireless Location: Challenges faced in developing techniques for accurate wireless location information", IEEE Signal Processing Mag., pp 24-40, July 2005.

[2] P.Khadivi, T.D.Todd, S.Samavi, H. Saidi, D.Zhao, "Mobile Ad Hoc Relaying for Upward Vertical Handoff in Hybrid WLAN/Cellular Systems", Journal of Ad Hoc Networks, Vol. 6, No. 2, pp 307-324, April 2008.

[3] P.Khadivi, S.Samavi, H.Saidi, "Restricted Shortest Path Routing with Concave Costs", Proc. of the 4th ACSIIEEE Int. Can/. on Computer Systems and Applications, UAE, 2006.

[4] S.D.Chitte et al. "Distance Estimation From Received Signal Strength Under Log-Normal Shadowing: Bias and Variance", Signal Processing Letters, Vol. 16, No. 3, pp 216-218,2009.

[5] K.Yu Y.J.Guo, "Anchor-free localisation algorithm and performance analysis in wireless sensor networks", lET Commun., Vol. 3, Iss. 4, pp. 549-560, 2009.

[6] L.Doherty et al. "Convex position estimation in wireless sensor networks", INFOCOM 2001, pp 1655-1663,2001.

[7] K.Chen et al. "A Localization Scheme for Underwater Wireless Sensor Networks", Int. Journal of Advanced Science and Technology, Vol. 4, pp. 9-16, March 2009.

[8] F.Kuroki et aI., "Distance Estimation at 60 GHz Band", IEEE A&E Systems Mag., pp. 10-12, June 2008.

[9] H.Koorapaty, "Barankin bounds for position estimation using received signal strength measurements", Proc. of VTC Can!, vol. 5, pp.2686-2690, 2004.

[10] A.J. Weiss, "On the Accuracy of a Cellular Location System Based on RSS Measurements", IEEE Trans. on Vehicular Technology, Vol. 52, NO. 6, pp. 1508-1518, Nov. 2003.

[11] A.H.Zahran and B.Liang, "Performance Evaluation Framework for Vertical Handoff Algorithms in Heterogeneous Networks", Proc. of the IEEE Int. Can! on Communications (ICC), Vol. 1, Pp 173-178, 2005.

[12] P.Khadivi, S.Samavi, H.Saidi, T.D.Todd, "Handoff in Hybrid Wireless Networks based on Self-Organization", Froc. of IEEE Int. Can/. on Communications (ICC), June 2006.