ESTIMATING POSITION OF MOBILE TERMINALS FROM DELAY ...

ESTIMATING POSITION OF MOBILE TERMINALS FROM DELAY MEASUREMENTS WITH SURVEY DATA M. McGuire, K.N. Plataniotis, A.N. Venetsanopoulos Department of Electrical and Computer Engineering, University of Toronto 10 King’s College Road, Toronto, ON M5S 3G4 Canada ABSTRACT An accurate method to locate mobile terminals is to measure the radio propagation time from the base stations to the mobile terminal. From these measurements, the distance from the base stations to the mobile terminal are estimated and an estimate of the location for the mobile terminal can be calculated. This method is highly accurate when the shortest distance propagation paths between the mobile terminal and base stations are not blocked. Unfortunately, when these paths are blocked the propagation delays are non-linear functions of the mobile terminal location and the accuracy of the standard location solution is reduced. This paper demonstrates how the conditional density of the location given measured propagation delays can be approximated as a sum of kernel density functions based on data collected during propagation surveys. These approximate density functions allow an accurate location estimate for the mobile terminal to be calculated from propagation delay measurements even with obstructed radio propagation. 1. INTRODUCTION The market for wireless networking services is undergoing fast growth. This growth is expected to continue with the proliferation of wireless data and digital multimedia devices. Increasingly, individuals are using portable wireless radio devices to access data as well as for voice communication. A growing concern is the ability to locate individuals making E911 calls with cellular telephones. The FCC in the United States has mandated that cellular network providers must be able to provide an estimated location of terminals making E911 calls that is accurate to within 100 meters for 67% of calls for network-based solutions[1]. For proposed third and fourth generation cellular networks, it is envisioned that wireless networks will be required to provide higher bandwidth multimedia data with strict Quality of Service requirements. It has been argued This work is partially funded by the Nortel Institute for Telecommunications

that one method to provide these services is to use mobile terminal location and prediction to allocate resources to the terminals[2]. Thus, mobile terminal location estimation could will become an integral part of wireless network management systems. There are other applications such as vehicular fleet management and location sensitive web-browsing which can provide new sources of income for cellular network providers using location estimation technology in the near future[3]. Several methods have been proposed in the literature for the location of mobile terminals in wireless networks based on Angle of Arrival (AoA), Time of Arrival (ToA), or Received Signal Strength (RSS) measurements[4]. Other options explored in the literature include adding GPS receiver hardware to the mobile terminals[5]. GPS can offer very high precision geo-location. This technology has the disadvantages that older mobile terminals can not be located with this technique and GPS does not work inside buildings or in areas where buildings or hills can block the LOS path to GPS satellites[6]. This paper will discuss the location of the mobile terminals based on ToA measurements of the mobile terminal signal at the fixed location base stations. This method is proposed for CDMA and GSM networks since these networks have modulation and multiple access schemes that allow the propagation delay to be measured[7, 8]. The propagation delays between the mobile terminal and fixed base stations are clearly functions of the mobile terminal’s location. If the conditional density functions of the propagation delays given the location of the mobile terminal were known then an estimate of the location could be easily calculated from delay measurements. In practice, these conditional density functions are not known. The best that can be done is to obtain estimates of the propagation delays at fixed locations within the propagation environment. These estimates can be obtained from field surveys or computer models of the propagation environment[9]. With this data there are two approaches that can be taken to construct estimates of the conditional density functions: Parametric and Non-Parametric techniques[10].

Most of the previous work on ToA location estimation have used the parametric density estimation technique. All propagation is assumed to be non-obstructed, or Line of Sight(LOS), and the measurement errors are assumed to be Gaussian. This technique has worked well in ‘simple’ propagation environment with few obstacles but does not perform well in complex environments such as urban microcells that have several obstructions[7]. A difficult problem is that Non Line of Sight (NLOS) propagation, where the shortest distance path between transmitter and receiver is blocked causes these estimators to give marginal results. This paper describes a non-parametric method to locate the mobile terminals. Similar techniques have been proposed for RSS location of mobile terminals using survey points[11]. The next section of this paper will discuss the estimation technique. Section 3 will describe the simulations used to evaluate the technique. Section 4 will give the results. Section 5 will list the conclusions. 2. ESTIMATION TECHNIQUE When a mobile terminal is to be located using ToA measurements,the data available is a set of ToA measurements from d base stations, Z, and a set of survey data for the area that the mobile terminal is known to be residing in. This area is identified by the hand off algorithm of the wireless network. The survey data can be described as a set of locations, θ1 , θ2 , ..., θn , and ToA vectors Z1 , Z2 , ... Zn where Zj is a vector of measured distance values in metres taken when a mobile terminal is at location θj . The measured ToA values are converted to distance estimates. These values are modeled as Z = c · g(θ) + V,

(1)

where g(θ) is a vector function which gives the true propagation delay for locations θ, c is the speed of light, and V is a random error vector that is independent of the location θ. The length of the Z vectors is d, the number of base station ToA measurements used to locate the mobile terminal. V is modeled as a d-variate Gaussian with zero mean and a covariance matrix of σ 2 multiplied by the d × d identity matrix. Given the set of ToA survey data, {Z1 , Z2 , ...Zn } and associated locations {θ1 , θ2 , ...θn }, we must find the location θˆ of a mobile which has a measured ToA-based distance vector Z. Note that both the measured ToA vector and the survey ToA vectors are contaminated with random measurement noise. The traditional approach to location estimation is to use the parametric Maximum Likelihood Estimator (MLE). All propagation is assumed to be LOS and thus the distance between the base station and mobile terminal is the mean mea-

sured distance. If the measurement noise is Gaussian then the MLE are the values of θˆ = (ˆ x, yˆ) which minimize the following likelihood expression: L=

d  2 X p Zk − (ˆ x − xk )2 + (ˆ y − y k )2 ,

(2)

k=1

where Zk is the measurement from base station k, and (xk , yk ) is the location of base station k. An additional enhancement that has been made is to note that NLOS propagation only causes the distance measurements to be increased so constraining solutions of (2) to values where p x − xk )2 + (ˆ y − yk )2 > 0 ∀k ∈ {1...d}. (3) Zk − (ˆ

This gives solutions that are more robust to NLOS propagation[7]. We will call this the Constrained Iterative MLE estimator below since an iterative procedure is used to solve (2) with constraint (3) on the solutions. The non-parametric MLE (assuming Gaussian measurement noise density), θˆM LE , can be calculated using ˆ j k2 kZ − Z → θˆM LE

ˆ i k2 ∀i ∈ {1, 2, ..., n} = min kZ − Z ≈ θj .

(4)

The approximation improving as n → ∞. The technique will be referred to as the MLE method below. The MLE has two short comings. First, the MLE can only return estimates of the mobile terminal location equal to one of the survey points. Second, the MLE makes limited use of information from survey points other than the survey point with the measured ToA value closest to the measured signal. To overcome these limitations other non-parametric estimators are proposed. It can be shown that the Minimum Mean Square Error (MMSE) or minimum variance of error estimate for unknown parameter θ given measurements Z is given by[12]: R Z θfΘ,Z (θ, Z) dθ ˆ θ = E [θ|Z] = θfΘ|Z (θ|Z) dθ = SR dθ S S fΘ,Z (θ, Z) (5) where S is the area in which the mobile terminal is known to reside. The problem at this point is that the densities in (5) are not known. The solution is to approximate the joint density as a sum of kernel functions[13]: fˆΘ,Z (θ, z) =     n z − Zj 1X θ − θj (hz )−d (hθ )−2 KZ Kθ . (6) n j=1 hz hθ The constants hz and hθ are smoothing parameters that determine the width or bandwidth of each of the kernel functions. For simplicity one usually chooses kernel functions

with the properties[10]: (a) K(w) ≥ 0 ∀ w ∈ Rd R (b) RRd K(w)dw = 1 (c) wK(w)dw = 0 Rd where d is the dimension of the kernel. Obviously, the kernel function is an d-variate density function for random variables of zero mean. If the estimated density function from (6) is substituted into(5), the result is     R Pn Z−Zj θ−θj −d −2 θ (h ) (h ) K K dθ z θ Z θ j=1 hz hθ S     θˆ = R P . n −d −2 θ−θj Z−Zj dθ K (h ) (h ) K θ z θ Z j=1 hz hθ S (7) If we assume that Kθ (·) satisfies the properties (b) and (c) above then this simplifies to the expression:   Pn −d Z−Zj θ (h ) K j z Z j=1 hz   θˆ = P (8) Z−Zj n −d (h ) K z Z j=1 hz This creates an estimator of the location using a non-parametric estimate of the density of the distance values which has the form[13] θˆ =

n X

θj w(Z, Zj ),

(9)

j=1

where n is the number of survey points used, and w(Z, Zj ) is a weight function. The weights are written in terms of kernel functions so that the weights are defined as: K(Z − Zj ) w(Z, Zj ) = Pn j=1 K(Z − Zj )

(10)

where K(x) is the user selected kernel function for the distance density, KZ (x/hz ). The kernel functions used in this paper are listed in Table 1. The user must select the value of hZ to be used. The optimal value of h depends on the value of σ 2 , i.e. the variance of the measurement noise, and the kernel function being used. An estimate of σ can be easily obtained by the base station. It will be shown in Section 4, how a good value of h can be selected and the accuracy is relatively insensitive to moderate deviations from the optimal value. In field implementations, real time performance of the algorithm is a major concern. One can take advantage of the property of the kernel functions in Table 1 that K(x) rapidly goes to zero as x moves away from the origin. This motivates the optimization that one can speed up the location estimation by only using the N survey points that have distance vectors closest to the measured distance vector Z. All the kernels degenerate to the MLE when N = 1. In the absence of any other information, we use the N survey

points which have the smallest squared Euclidean distances, kZ − Zj k2 , from the measured distance vector. The optimal value of N is determined by the magnitudes and number of discontinuities in the unknown function Z(θ). Lower values of N can be used if Z(θ) has several large discontinuities. kernel name Parzen Gaussian[14] Parzen Laplace[14] Distance based[15]

kernel function K(x)   2 1 √ − kxk d exp 2 ( 2π)
1 1 2 exp −kxk Qd

k=1

1 Kp 1+(x p k)

kxkp isR the Lp distance of x from the origin. ∞ 1 1 Kp = −∞ 1+xp dx. Table 1. Kernel functions

3. DESCRIPTION OF SIMULATIONS The location estimation methods were evaluated using simulations. A regular Manhattan street microcell model was considered with dimensions and propagation characteristics as described in[7]. The environment is shown in Figure 1. The hatched areas represent buildings. When the LOS path between a base station and a mobile terminal is unobstructed, the propagation distance is simply the Euclidean distance between the base station and mobile terminal location. In the Non Line of Sight (NLOS) case, when the the LOS path is blocked, it is assumed that the shortest propagation path between the mobile terminal and base station is via diffraction around a building corner. An example of this propagation path is shown in Figure 1 where the propagation distance would be dc + dr . We assume that the cellular network implements the ideal location based hand off algorithm; the mobile terminal communicates with the base station that is closest to it. This makes the the region that the mobile terminal resides in while communicating with the central base station the region bounded by the dashed line. The central base station and two other base stations that have the lowest ToA values are used to locate the mobile terminal. That is, the base station that is serving the mobile terminals and the two other base stations that have the lowest measured propagation delays from the mobile terminal are used to estimate the mobile terminal location. This base station selection method was selected since it is consistent with how the base stations would be selected in field implementations of the algorithm.

Others [7] have assumed that the closest base stations are used. This method was not chosen since it injects an amount of side information into the location estimation process that would not be present in a field implementation of the algorithm. When the closest base stations are used then the presence and exclusion of base stations in the measurement set allows for deterministic removal of areas from S. For example, if we know that base station 1 is closer than base station 2 than all regions in S that are closer to base station 2 than base station 1 can be removed from S. With the measurement based algorithm no such deterministic partitioning is possible. It is possible with our method of base station selection, theoretically, to calculate a probability for any location in S that the set of base stations making measurements would be selected. These probability values could be used to weight the different survey points and improve the location estimate accuracy. Unfortunately, this calculation is difficult and is not implemented in this paper but it is a topic for future research. The value of σ is varied to see how well the location algorithms perform with different levels of measurement noise. Random locations for the mobile terminal are generated by sampling from a uniform distribution over street locations within the diamond shaped region in Figure 1. The propagation environment is dominated by locations within buildings. The areas of most interest to cellular network operators, however, are street locations. Since most E911 calls made from cell phones will be made from street locations evaluation of location error can give greater weight to street locations[1]. Simulations were first performed with a hundred survey points, n = 100, randomly generated from a uniform distribution over the region described above. In the field, the survey is more likely to be performed in some regular manner with measurement made in some regular pattern. Sets of simulations were performed to evaluate how well the estimators worked when survey points with non-random locations are used. Survey points were placed every 12 metres on the line running in the middle of the vertically aligned street and every 12 metres on the line running in the middle of the horizontally aligned street for a total of 100 points. 4. RESULTS The first set of simulations show the accuracy of the estimators for different values of h. Figure 2 shows the results for N = 100, all survey points are used for estimation, and Figure 3 shows the results for N = 10, when only the 10 survey points with delay measurements closest to the delay measurement for the mobile terminal are used. In both cases, the standard deviation of the noise is 15 metres. The Gaussian Kernel gave the best performance with

Fig. 1. Manhattan Propagation Environment h = 2σ. Using N = 10 made the estimator more robust to variations in the used h value in that the estimator accuracy is not as greatly reduced by variations of h off of the optimal value. The other Kernels give location estimates with only slightly less accuracy than the Gaussian Kernel. For the simulations given below N = 10 and h = 2σ. The robustness of the non-parametric estimators to variations of the measurement noise were tested next. These results are presented in Figure 4. The kernel based estimators all perform better than the MLE estimator with performance gap increasing as the variance of the measurement noise increases. Another set of simulations were performed to see if using a survey with deterministic survey point locations reduced the efficiency of the estimators. The results are shown in Figure 5. The last set of simulations was done to compare the accuracy of the non-parametric estimator described above with the parametric estimators. The results are shown in Figure 6. As can be seen the non-parametric location estimator always performed better. The literature on these parametric estimators has shown that better results can be obtained if the NLOS base stations can be identified[7] but the techniques for performing this identification are not specified. In any case the reported results are still worse than those obtained for the non-parametric estimator shown here. 5. CONCLUSIONS It has been demonstrated above that a non-parametric kernelbased location estimation estimation algorithm gives location estimates with higher accuracy than parametric MLE location estimation methods for a cellular radio network

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Fig. 5. Estimator performance for differing σ values and deterministic survey point locations(N = 10).

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Fig. 3. Estimator performance for differing h values (N = 10, h = 15 metres).

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Fig. 6. Estimator Comparison 15 15

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with small radius cells that have many obstructions to LOS propagation. This non-parametric technique requires survey samples of delay measurements taken at known locations to generate approximate conditional density functions of location given a delay measurement. This efficiency of this estimator is dependent on the value of a smoothing parameter or kernel size. It was shown that the number of survey points needed for a good estimation of location is low. A good value of the smoothing parameter can be easily estimated. Furthermore, by only using a subset of the survey points that have delay measurements close to the measured delay for the mobile terminal the location estimate can be made very robust to errors in the smoothing parameter value. 6. REFERENCES [1] FCC, “OET bulletin no. 71, guidelines for testing and verifying the accuracy of wireless E911 location systems,” April 2000. [2] Z.J. Haas, J.H. Winter, and D.S. Johnson, “Simulation results on the capacity of cellular systems,” IEEE Transactions on Vehicular Technology, vol. 46, no. 4, pp. 805–817, November 1997. [3] T. Lewis, “Tracking the ‘anywhere anytime’ inflection point,” IEEE Computer, vol. 32, no. 2, pp. 134–136, February 2000. [4] I. Jami, M. Ali, and R.F. Ormondroyd, “Comparison of methods of locating and tracking cellular mobiles,” in IEE Colloquium on Novel Methods of Location and Tracking of Cellular Mobiles and Their System Applications, 1999, pp. 1/1–1/6. [5] M.J. Meyer, T. Jacobson, M.E. Palamara, E.A. Kidwell, R.E. Richton, and G. Vannucci, “Wireless enhanced 9-1-1 service – making it a reality,” Bell Labs Technical Journal, vol. 1, no. 2, pp. 188–201, 1996. [6] N. Bulusu, J. Hedemann, and D. Estrin, “GPS-less low-cost outdoor localization for very small devices,” IEEE Personal Communications, vol. 7, no. 5, pp. 28– 34, October 2000. [7] J.J. Caffery, Jr and G.L. St¨uber, “Overview of radiolocation in CDMA cellular systems,” IEEE Communications Magazine, vol. 36, no. 4, pp. 38–45, April 1998. [8] J. Winter and C. Wengerter, “High resolution estimation of the time of arrival for GSM location,” in IEEE Spring Vehicular Technology Conference, 2000, pp. 1343–1347.

[9] R.B. Ertel, P. Cardieri, K.W. Sowerby, T.S. Rappaport, and J.H. Reed, “Overview of spatial channel models for antenna array communication systems,” IEEE Personal Communications, pp. 10–22, February 1998. [10] D.W. Scott, Multivariate Density Estimation: Theory, Practice, and Visualization, John Wiley & Sons, Toronto, 1992. [11] Z. Salcic and E. Chan, “Mobile station positioning using GSM cellular phone and artificial neural networks,” Wireless Personal Communications, vol. 14, no. 3, pp. 235–254, 2000. [12] A.P. Sage and J.L. Melsa, Estimation Theory with Applications to Communication and Control, R.E. Kreiger Publishing Co., New York, 1979. [13] K.N Plataniotis, D. Androutsos, S. Vinayagamoorthy, and A.N. Venetsanopoulos, “Color image processing using adaptive multichannel filters,” IEEE Transactions on Image Processing, vol. 6, no. 7, pp. 933–949, July 1997. [14] G.A. Babich and O.I. Camps, “Weighted parzen windows for pattern classification,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no. 5, pp. 567–570, May 1996. [15] M. Barni, V. Cappellini, and A. Mecocci, “A modified metric to compare distances,” Pattern Recognition, vol. 25, no. 5, pp. 667–677, May 1992.