2 Conference on Information Sciences an Systems, The Johns Hopkins University, March 2, 2 Time-of-Arrival Estimation in Non-Line-Of-Sight Environments Sinan Gezici, Hisashi Kobayashi an H. Vincent Poor Department of Electrical Engineering Princeton University Princeton, NJ 8 {sgezici,hisashi,poor}@princeton.eu Serar Yüksel an Tamer Başar 2 Coorinate Science Laboratory University of Illinois at Urbana-Champaign 8 West Main Street, Urbana, IL {yuksel,tbasar}@ecision.csl.uiuc.eu Abstract A new estimator is propose to extract the true istance between a mobile an a base station from a set of time-of-arrival (TOA) ata, corrupte by unknown non-line-of-sight (NLOS) errors an Gaussian measurement noise. Characteristics of the estimator are iscusse for a class of NLOS errors with variance consierably greater than that of the measurement noise, which is usually the case in practice. A quantization approach is use to estimate the probability ensity function of the observe TOA measurements an the TOA estimator is compute from this ensity estimate. Keywors: Wireless location, NLOS error, parameter estimation, quantization. I. Introuction In the stuy of mobile positioning in wireless communication systems, accurate location estimation of a mobile station has proven to be an important problem. Multipath, non-lineof-sight (NLOS) propagation an multiple access interference are often the main sources of errors in geolocation, an make mobile positioning challenging. Among these error sources, NLOS is usually consiere to be the most crucial one an several practical algorithms to reuce the effects of this error have been propose in the literature. In [2] a hypothesis testing approach is use to etermine NLOS base stations an then the line-if-sight (LOS) reconstruction is achieve using the history of the measurements base on the hypothesis testing results. In [] statistics of the measurement noise are use to estimate the istances between the mobile an the base stations. However, the assumption mae that measurements lower than the true istance are ue to the measurement noise only may not hol in general. In [] an algorithm is use to mitigate the NLOS errors when no prior information is available assuming that there are more than three base stations for geolocation purpose. In this paper, a time of arrival (TOA) estimator is propose for a class of NLOS error statistics. It is shown that when the NLOS error has a much larger variance than the Gaussian measurement error, which is usually the case [], an when its probability ensity function (pf) is approximately constant aroun the origin, then the first erivative of the pf of the measurements has a peak near the true TOA. This observation is the key to the estimator propose in this paper. The remainer of this paper is organize as follows. Section II escribes the estimator an iscusses its properties. Use of This research is supporte in part by the U.S. National Science Founation uner grant CCR-99-79, an in part by the New Jersey Center for Wireless Telecommunications. 2 This research is supporte in part by the U.S. National Science Founation uner grants CCR--897 an ECS 2-28. the estimator in a geolocation problem is aresse in Section III, followe by simulation results in Section IV. Finally, Section V presents some concluing remarks. II. Time of arrival estimation The estimator we propose is base on the estimation of the pf of the observe set of TOA measurements, which are the outcomes of a ranom process with the following characteristics: Y + E + N () where is the LOS elay to be estimate, N is the measurement error, which is moelle as a Gaussian ranom variable with zero mean an known variance σ 2, an E is the NLOS incurre elay taking a value between an some value L, possibly infinite. Define Z + E. Note that the pf of Z will be zero for Z, an the pf of the observation Y will be the convolution of the pf of Z an the pf of the Gaussian noise, N. Depening on the properties of the pf of E, we can make the following property. Property 2. If the ensity of the NLOS error E is uniform with a variance much larger than the variance of the Gaussian noise, then the probability ensity of Y has its secon erivative equal to zero (sale point) at almost the true timeof-arrival,. Furthermore, the assertion remains vali for a non-uniform NLOS error with boune pf, that is, f E (x) B for all x, if the pf of the NLOS error R is almost constant between an α, where α < L satisfies α ( x2 /σ 2 )e x 2 2σ 2 x /B an is consierably larger than the stanar eviation of the Gaussian measurement noise, σ. An argument supporting this property is as follows. Since Z + E, the pf of Y at a point t is f Y (t) Z +L f Z (τ)f N (t τ)τ, (2) where f Z is the pf of Z an f N is the ensity associate with the Gaussian noise. Taking the erivative with respect to t, we have f Y (t) t Z +L f Z (τ) fn (t τ) τ. () t Setting the secon erivative to zero to fin the point that achieves the maximum increment in the ensity of Y, we obtain Since 2 f Y (t) t 2 Z +L 2 f N (t τ) t 2 f Z(τ) 2 f N (t τ) t 2 τ. () (t τ)2 e (t τ)2 σ 2 2σ 2, ()
f W (x)/x.... 2 x 2 2 2 8 x Figure : First erivative of the sum of the NLOS error an the Gaussian error for L m an σ 2 2m. f 2 W (x)/x2.... 2 x 2 2 2 8 x Figure 2: Secon erivative of the sum of the NLOS error an the Gaussian error for L m an σ 2 2m. equation () leas to Z +L Z +L f Z(τ) σ 2 e (t τ)2 2σ 2 τ f Z(τ) (t τ) 2 e (t τ)2 2σ 2 τ. () If the NLOS error E has a uniform istribution, the associate pf can be taken out of the integral an cancelle. Furthermore, if the Gaussian noise has almost all of its power between L an L, then at t the left han sie is equal to σ2 which 2 is equal to the right-han sie. Thus, for sufficiently large L, the secon erivative of the pf of the observe ranom variable is approximately zero at true TOA,. This point is the first (an only, if the NLOS is uniform) positive peak of the first erivative of the observe pf. The fact that the secon erivative is close zero at means that the true TOA is close to the sale point since the secon erivative of the pf of the measurements changes consierably aroun that point (see Figure 2 where ), which can be seen from the non-negligible negativity of the thir erivative at t : f Y (t) t Z +L (τ ) f Z (τ) (τ )2 σ 2 e (τ )2 2σ 2 τ. If the NLOS istribution is not uniform, then the arguments above will not hol in general, especially when the measurement noise variance is comparable to the NLOS error variance. However, in practice, as is argue in [] an [2], the measurement noise variance is usually much smaller than the NLOS error variance, an the pf of the NLOS error can be regare as practically constant for an interval comparable to σ an the above result still hols approximately as shown below. After a change of variables an using f E(x) f Z(x + ), the expression () at t can be expresse as Z L f E(x) ( x2 /σ 2 )e x2 2σ 2 x. (7) If we consier the integrals from to α an α to L separately, we have approximately + f E () Z L α Z α ( x2 /σ 2 )e x2 2σ 2 x f E (x) ( x2 /σ 2 )e x2 2σ 2 x. (8) Due to the conitions in the proposition, the first term in (8) is approximately zero. Also since L is very large compare to σ, R L ( x 2 /σ 2 )e x2 2σ 2 x is approximately R L zero. Therefore, ( α x2 /σ 2 )e x 2 2σ 2 x /B as well. Hence, the secon term is also approximately zero. Therefore, when the pf of the NLOS error is almost constant aroun the origin an has much larger variance than the measurement noise variance, the true TOA will be very close to the point where the secon erivative is zero, an might be assume to be equal to that sale point without much loss of performance. As a worst limiting case, where the assumption regaring the relationship between the variances completely fails, the NLOS error ensity is impulsive aroun the origin, meaning that there are only LOS observations. In this case the secon erivative will be zero at the point σ. Thus, in the worst case for a ecreasing ensity of NLOS, the error in the estimation will be σ. Actually, in this case, not the secon erivative but the first erivative being equal to zero characterizes the true TOA. However, it is usually possible to istinguish LOS measurements from the NLOS measurements by some ecision criterion []. Hence the LOS case can be consiere separately. The estimator is characterize by the pf of the observe ranom variable as follows: A pf estimation algorithm is employe, for which a histogram approach is use, which correspons to interpreting the observe ata as the input to a uniform quantizer with a specifie bin size. The number of observations per quantization bin is use to compute the probability mass function (pmf) of the ata. The smaller bin ege value corresponing to the quantizer bin having the first local peak in the ifference between the current bin an the preceing bin is the estimator output. Note that the choice of the bin size of the quantizer is an important parameter. The estimator can be formulate as follows. Given m inepenent ientically istribute (ii) measurements Y,..., Y m,
we quantize these measurements into N b bins. Assume that the ith bin extens from (i )l b to il b where l b is the bin size. Let S,..., S NB enote the number of samples in the bins an D i S i S i the ifference in the number of samples between ajacent bins. Then, the inex of the bin incluing the true TOA is estimate as ˆb arg min i {D i D i > an D i+ D i }, (9) an (i )l b gives the TOA estimation. Note that it is not the reconstruction value but the bin ege that gives the true TOA estimation, since the bin ege characterizes the increment better. With this efinition of the estimator, we have the following property: Property 2.2 Assume that the NLOS error is uniformly istribute with support much larger than the Gaussian measurement error stanar eviation. Then, as the number of samples goes to infinity, the error of the estimator is almost surely confine to the interval [-. l b,. l b ] for small values of l b. Let m enote the number of samples. As m, S i/m p i almost surely, where p i Z ilb (i )l b f Y (y)y, () with Y given by (). Thus, (D i D i )/m p i 2p i +p i 2 almost surely. Let the kth bin contain the true TOA. From the above analysis of the pf of the measurements, we know that the pf increases until some point with increasing first erivative an after almost the true TOA it continues to increase with a ecreasing first erivative. In other wors, the pf of the measurements is strictly convex before the true TOA an strictly concave after that for some time. Consier D k+ D k. For large m, we have (D k+ D k )/m p k+ + p k 2p k Z (k+)lb Z (k )lb Z klb When the support of the uniform error is much larger than the stanar eviation of the Gaussian measurement error an the bin size is sufficiently small, the value of the integral (2) is approximately zero for l b /2. From the convextiy/concavity of the pf aroun the origin, we also have for < l b /2, D k+ D k < R an for > l b /2, D k+ D k >. (k i)l Since D i D i b [f (k i+)l W (y + l) + f W (y l) b 2f W (y)]y, the convexity of f W (.) to the left of the origin an concavity of it to the right of the origin to some extent together with the previous result implies that D i D i > for i k, k,... an D i D i < for i k + 2, k +,... Therefore, the only bins that can satisfy the conitions of the estimator (9) are the kth or the (k + )st bins: For < l b /2, the kth bin an for > l b /2, the (k + )st bin satisfies the conitions of the estimator. Therefore, for < l b /2, the TOA estimate is (k )l b an for > l b /2, it is kl b. Since (k )l b +, the error is between.l b an.l b. Note that for pf s satisfying the alternative conitions state in Property 2., Property 2.2 is still approximately true. The histogram can be interprete as the quantization of the erivative of the pf of the measurements. In this case, the increments can approximately be represente by f Y ((i )l b ) lb 2 p i p i for small bin sizes. Thus the erivative is represente by rectangular bins, an the bin ege of the quantizer can be regare as the representative sample for the erivative. The erivative of the ensity function of the total error with a uniform NLOS noise is epicte in Figure. As is seen, the erivative function goes to zero after the peak, an then has a negative peak. This figure can be use to introuce a boun on the bin size for the estimator. The estimator will compare the competing erivatives for the peak an output the peak as the estimate bin. The bins aroun the upper part of the quantizer will be the caniates for the maximum value, an if those bins o not interfere with the lower portion (negative values) of the erivative curve, the estimation outcome will be at most half the quantizer bin size away from the true TOA (which is in the figure). Assuming symmetry in the upper part of the curve with respective to the peak of the erivative, the bin sizes less than the ifference between the peak an the point where the erivative crosses zero, will cause a istortion not greater than l b 2, whereas larger bin sizes may result in interference with the values in the negative portion an cause an error. The ifference between the sale point (peak of the erivative) of the ensity of the observation an the maximum point (where the erivative is zero) is on the orer of a few stanar eviations of the noise. Thus the quantizer bin size shoul be upper boune by a few σ s. f Y (y)y + f Y (y)y 2 f Y (y)y kl b (k 2)l b (k )l b III. Mobile Location Z klb Location of a mobile in a wireless communications system [f Y (y + l b ) + f Y (y l b ) 2f Y (y)]y. () can be etermine from TOA measurements from a number of (k )l b base stations (BSs). Traitionally, the location of the mobile Since the true TOA is in the kth bin, we can express as is estimate by a least squares (LS) estimator [], which is (k )l b + where l b. Defining W N + E as the optimal estimator when the mobile is LOS to all the BS s. the sum of the NLOS an measurement errors an using the When the NLOS error is not known, this approach is still fact that f W (x) f Y (x + ), the last integral can be written employe an the NLOS observations are given less weight in as: orer to mitigate the effect of the NLOS error. Z lb If the locations of the mobile an the BS s are enote [f W (y + l b ) + f W (y l b ) 2f W (y)]y. (2) by p m an p,..., p b respectively with b being the number of BS s, the location estimate is given by the p that minimizes bx i w i ( p p i vy i ) 2, () where. enotes L2-norm, v is the spee of light, Y i is the TOA measurement from the ith BS an w i is the weight that reflects the reliability of the measurement from the ith BS, which might be consiere to be inversely proportional to the variance of the measurements. In NLOS situations, even though the NLOS BS is given smaller weight, the error will still be significant unless there
. m m m2 m 7. Our Algorithm RRE Algorithm...... 2. 7 8 9 2 Bin Size (m) 2. 2 2 Number of Samples Figure : Mean Absolute Error versus bin size for ifferent sample sizes for the uniform NLOS error, [, m], an Gaussian measurement error with variance 2m 2. Figure : Mean Absolute Error for ifferent algorithms for the uniform NLOS error, [, m], an Gaussian measurement error with variance 2m 2. are at least three LOS BS s (for two-imensional location). Therefore, we suggest employing our TOA estimation technique for the NLOS BS s before using the LS estimator. We assume that the NLOS BS s can be etermine from the measurements using a variance test []. Consier a series of TOA measurements from a NLOS BS. To apply the TOA estimation technique, consier a winow aroun the specific TOA measurement an take those measurements as input to the estimator. Note that the samples shoul be sufficiently space in time so that they are inepenent. Then, for NLOS BS s Y i in equation () is replace by the result of the TOA estimation an the final location is estimate by the LS scheme. IV. Simulation Results Consier the moel in () where the Gaussian measurement error has variance σ 2 an the NLOS error is uniform between an L. Assume that m ii TOA measurements are taken accoring to this moel an the location of the mobile oes not change consierably uring these measurements. This assumption means that the NLOS error is assume to be inepenent from sampling instant to sampling instant, which is ue to the ynamic environment an/or the movement of the mobile, but the mobile s location can be consiere to be constant for the purposes of TOA estimation. These two seemingly inconsistent assumptions can be reconcile since the NLOS error variations are small-scale variations while the mobile s location is a large-scale property. For the simulations, the observe TOA ata have been converte to istance measurements for convenience. The variance of the Gaussian measurement error is taken to be 2 m 2. In Figure, the NLOS error is taken to be uniformly istribute between an m an the mean absolute error (MAE) is calculate for ifferent bin sizes, l b. For each bin size, 2, trials were performe with the true istance uniformly istribute between some range so that the probability that the true istance is at any location in a bin is always uniformly istribute. From the figure, it is seen that there is an optimal bin size for each sample size, m. For small bin sizes, number of samples may not be sufficient to represent the erivative value correctly resulting in higher errors. For large bin sizes, even though the correct bin can be chosen, the estimation value may not be very close to the true istance ue to the large size of the bin. Also note that when the sample size increases, the optimal value of bin size ecreases. Figure compares the performance of the estimator to the Robust Range Estimation (RRE) algorithm []. The error of the RRE algorithm is higher than the error of our estimator since the assumption use in the RRE algorithm, namely that the measurements lower than the true istance comes from the Gaussian error assumption, may cause some small errors. However, that algorithm oes not require specific properties of the NLOS pf which are assume for our estimator. But in practical situations the conitions on the pf are not so tight an are satisfie by most NLOS error istributions. In Figure, the NLOS error is taken to be exponentially istribute with mean 2 m an the mean absolute error (MAE) is calculate for ifferent bin sizes. For each bin size, 2, trials were performe with the probability that the true istance is at any location in a bin again uniformly istribute. From the figure, the same observations as the uniform NLOS error case can be mae. Also the algorithm clearly works for the exponential NLOS error as well since the variance of the NLOS error is consierably larger than the Gaussian measurement error variance. Figure compares the two estimators. Again the algorithm propose in this paper achieves lower error values. However, in this case, the results are closer since the NLOS pf is exponentially istribute an Proposition 2. applies since it has a large variance compare to the Gaussian error. Now consier the mobile tracking scenario shown in Figure 7, in which there are three BS a at (, ), (, ) an (, 29.8) an the mobile moves from (, 8) to (2, 8) on the soli line shown with a velocity of m/s. Twenty TOA measurements are taken per secon. The first BS is assume to be NLOS with an exponential NLOS error with mean 2 m, an the other BS s are assume to be LOS. The Gaussian measurement error variance is 2 m 2 for all measurements. 2 samples aroun a given sample are consiere for the TOA estimation for the NLOS BS an a bin size of 2 m is use for the quantization. Also the weights are taken to be inversely proportional to the variances of the measurements for the LS
Mean Absolute Error (m)... m m m2 m 2. 7 8 9 2 Bin Size (m) Figure : Mean Absolute Error versus bin size for ifferent sample sizes for the exponential NLOS error with mean 2m an Gaussian measurement error with variance 2m 2... Our Algorithm RRE Algorithm 2. 2 2 Number of Samples Figure : Mean Absolute Error for ifferent algorithms for the exponential NLOS error with mean 2m an Gaussian measurement error with variance 2m 2. estimator. From the figure it is seen that employing our TOA technique for the NLOS BS improves the location accuracy. The MAE for the traitional LS algorithm is. m whereas it is 2.9 m for the moifie scheme. V. Conclusion In this paper we have introuce a practical algorithm to estimate true TOA of a signal between a mobile an a base station in the presence of NLOS noise with unknown statistics an Gaussian noise with a variance consierably smaller than that of the NLOS error. It has been shown that uner reasonable conitions, TOA estimation can be performe by locating the first peak of the first erivative of the pf of the measurements. A histogram approach can then be employe to approximate the pf of the measurements an thus obtain this estimate. The simulations are performe to asses the performance of the estimator. Our simulations show the superiority of this approach to existing methos. References [] M. P. Wylie-Green an S. S. Wang, Robust range estimation in the presence of the non-line-of-sight error, Proc. IEEE th Vehicular Technology Conference, Fall 2 vol., Atlantic City, NJ, pp. -. [2] M. P. Wylie an J. Holtzman, The non-line of sight problem in mobile location estimation, Proc. th IEEE International Conference on Universal Personal Communications, 99, Cambrige, MA, pp. 827-8. [] P. C. Chen, A non-line-of-sight error mitigation algorithm in location estimation, Proc. IEEE Wireless Communications an Networking Conference, 999, vol., New Orleans, LA, pp. -2. [] J. Caffery, Jr. an G. Stuber, Subscriber location in CDMA cellular networks, IEEE Trans. Vehicular Technology, vol. 7, pp. -, May 998. [] J. Borras, P. Hatrack an N. B. Manayam, Decision theoretical framework for NLOS ientification, Proc. IEEE Vehicular Technology Conference, Spring 998 vol. 2, Ottawa, Canaa, pp. 8-87. 2 BS 2 y (m) LS Moifie LS BS BS 2 2 2 x (m) Figure 7: Mobile Tracking Scenario. Mobile moves from left to right with velocity m/s an 2 TOA measurements are taken per secon. Only BS is NLOS.