Non-Parametric Non-Line-of-Sight Identification 1

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1 Non-Paraetric Non-Line-of-Sight Identification Sinan Gezici, Hisashi Kobayashi and H. Vincent Poor Departent of Electrical Engineering School of Engineering and Applied Science Princeton University, Princeton, NJ Abstract Recently, there has been uch interest in accurate deterination of obile user locations in cellular environents. A general approach to this geolocation proble is to gather tie-of-arrival easureents fro a nuber of base stations (BSs and to estiate user locations using the traditional least square approach. However, in non-line-of-sight (NLOS situations, easureents are significantly biased. Hence, very large errors in location estiation ay be introduced when traditional techniques are adopted. For this reason, before eploying an algorith for location estiation, it is useful to know which BS s are in line-of-sight (LOS and which are in NLOS of the obile station. In this paper, a non-paraetric approach to this NLOS identification proble is proposed. Since the statistics of NLOS errors are usually unknown, a non-paraetric probability density estiation technique is eployed to approxiate the distribution of the easureents. Then, an appropriate etric is used to deterine the distance between the distribution of the easureents and the distribution of the easureent noise. Depending on the closeness of the distributions, the propagation environent is classified as LOS or NLOS. In a situation where reliability of easureents fro a BS is to be quantified, the distance can be used to represent the reliability of the easureents as well as to classify the station. I. INTRODUCTION Recently, the subject of obile positioning in wireless counication systes has drawn considerable attention. With accurate location estiation, a variety of new applications and services such as Enhanced-9, location sensitive billing, iproved fraud detection, intelligent transport syste (ITS and iproved traffic anageent will becoe feasible [2]. Mobile positioning using radiolocation techniques usually involves tie of arrival (TOA, tie difference of arrival (TDOA, angle of arrival (AOA, or signal strength (SS easureents, or soe cobination of these ethods. Multipath, non-line-of-sight (NLOS propagation and ultiple access interference are often the ain sources of errors in geolocation, and ake obile positioning challenging. Aong these error sources, NLOS is perhaps the ost crucial one. In an NLOS situation, TOA easureents that are used to estiate the distance between a obile station (MS and a base station (BS are severely biased. In this case, using traditional location algoriths ay result in large errors in location estiation []. However, if it is known that the MS is in NLOS with respect to a BS, then soe special ethods can be applied This research was supported in part by the U.S. National Science Foundation under Grant CCR , and in part by the New Jersey Center for Wireless Telecounications. depending on the scenario. For exaple, if at least three BS s are in LOS of the MS, then the easureents fro the NLOS BS s can be discarded in obtaining a two-diensional location estiate. Alternatively, when the Recursive Weighting Algorith [2] is eployed to reduce NLOS propagation errors, knowledge of LOS/NLOS BS s becoes iportant. In other words, identification of NLOS BS s can help considerably to iprove the location estiation. The proble of NLOS identification is essentially a detection proble. It copares the LOS hypothesis to the NLOS hypothesis. The probability distribution of the easureents under the LOS hypothesis is usually known except for its ean. If the distribution under NLOS hypothesis is also assued to be known, then the proble can be solved by the conventional hypothesis testing ethod [3]. However, the probability distribution of NLOS errors, hence that of the easureents under the NLOS hypothesis, is usually unknown. Therefore, a technique which does not assue the knowledge of NLOS error statistics should be developed. In this paper, a non-paraetric NLOS identification technique is proposed. Since the statistics of TOA delays due to NLOS are not known exactly, a non-paraetric approach is adopted to approxiate the probability density function of the easureents. Then, a suitable distance etric between a known easureent error distribution and a nonparaetrically estiated distance easureent distribution is defined to deterine whether a given BS is within LOS or NLOS of the MS. The distance between these two distributions can also be used as a reliability easure for the easureents fro the given BS. The reainder of the paper is organized as follows. Section II forulates the proble of NLOS identification and describes the non-paraetric NLOS identification algorith. The perforance of the algorith is evaluated in Section III by siulation studies. Finally, soe concluding rearks are ade in Section IV. II. NON-PARAMETRIC NLOS IDENTIFICATION Consider a situation in which independent identically distributed (iid range easureents (obtained fro TOA easureents ultiplied by the speed of light between an MS and a BS are taken. Assue that the change in the location of the MS during these easureents can be ignored. Hence the distance between the MS and the BS can be considered approxiately constant for the geolocation purpose. Then, for /03/$ IEEE. 2544

2 the ith easureent, the hypotheses can be expressed as: : r i = d + n i H : r i = d + n i + e i, ( for i =,...,, where is the LOS hypothesis and H is the NLOS hypothesis. In the forer case, the easureent is odelled as the suation of the true distance d and a easureent noise, n i, while in the latter case, the NLOS error e i is also present, which is odelled by a positive rando variable. We assue that the easureent noise statistics are copletely known and is odelled by a zero ean Gaussian rando variable. However, neither d nor the probability density function of the NLOS error are known. Therefore, it is not possible to invoke conventional hypothesis testing techniques like generalized likelihood ratios. Let the probability density function (pdf of the easureent noise be p n (x, which is copletely known. Then, the pdf of the easureents in the LOS hypothesis case is given by p n (x d. Note that this distribution is copletely known except for one paraeter, d, which affects only the ean of the distribution. The ain idea in the non-paraetric NLOS identification test is to copare the closeness of this pdf to the pdf of range easureents. Thus we first approxiate the pdf of the range easureents non-paraetrically, copare the closeness of this pdf to the LOS pdf by defining a distance etric, and then decide LOS/NLOS after a threshold test. This test can be suarized as follows: Estiate the pdf of the distance fro iid range easureents. Let this estiate be denoted by ˆp r (x. 2 Calculate the distance between p n (x d and ˆp r (x for all possible d values and find the iniu distance. 3 Copare this iniu distance to a threshold: Decide if the iniu distance is saller than the threshold, and decide H otherwise. We will discuss these steps in ore details in the following subsections. A. Parzen Window Density Estiation In order to estiate the pdf of the distance, a non-paraetric density estiation technique, called Parzen window density estiation, is eployed, which approxiates the pdf using soe window functions around the saples. The reason for eploying this technique is its flexibility in choosing density estiation paraeters depending on the saple size. Given iid distance easureents r,..., r, the distance pdf can be estiated by the following forula [4] ˆp r (x = φ( x r i, (2 h h where φ(. is the window function and h is a scaling paraeter. The window function ust be a probability density function in order for ˆp r (x to be a valid pdf. In other words, it is always non-negative and integrates to one. Coonly used window functions include Gaussian and rectangular windows [4]. B. Distance Function After obtaining the approxiate pdf of the distance, our next step is to deterine whether these distance easureents are coing fro p n (x d or the pdf under the NLOS hypothesis. Since the pdf under the NLOS hypothesis is unknown, it is reasonable to copare the distance between p n (x d and ˆp r (x and accept the LOS hypothesis if the distance is saller than a threshold, that is, if the two distributions are sufficiently close. Since the true distance, d, is unknown, the iniu distance between p n (x d and ˆp r (x ust be calculated aong all possible d s. The Kullback-Leibler (KL distance [5] can be used to calculate the distance between two probability distributions. For given pdf s p and p 2, the KL distance between the is given by D(p p 2 = p (x log p (x dx. (3 p 2 (x C. Decision Criterion The decision criterion to deterine LOS or NLOS hypothesis becoes the following test: inf d {D(ˆp r (x p n (x d} δ, (4 H where δ is the threshold. If the value of the d iniizing the decision variable can be found, the test can be expressed siply as D(ˆp r (x p n (x ˆd δ. (5 H We assue that the easureent noise is a zero ean Gaussian rando variable, which is a valid approxiation when the TOA s are acquired with a atched filter approach at high signal-to-noise ratio (SNR [6]. Then, p n (x d is expressed as p n (x d = e (x d2 /(2σ 2. (6 2πσ In this case, the following result indicates the siplification of the decision test. Proposition 2. For a zero ean Gaussian easureent error and a syetric window function, i.e., φ(x =φ( x for all x, the value of d iniizing the distance function of (4 is the saple ean of the easureents, that is, ˆd = r i. Proof See Appendix A. Proposition 2. states that for a syetric window function, the iniu distance to be used in the decision criterion can be coputed by siply shifting the Gaussian easureent error pdf by the saple ean of the easureents and calculating the KL distance between this shifted pdf and the estiated pdf, ˆp r (x /03/$ IEEE. 2545

3 Another iportant issue is the appropriate choice of the threshold value, δ. Since the pdf s are not known exactly under either hypothesis, it does not see possible to set the false alar (i.e. isinterpret a LOS situation as NLOS and iss detection (i.e. isinterpret an NLOS situation as LOS probabilities. However, the following result states that in soe situations the false alar probability can be set even though the true distance d is not known, that is, without any inforation about the ean of the rando variable under. Proposition 2.2 For a zero ean Gaussian easureent error and a syetric window function, the false alar probability can be set independently of the true distance between the obile and the base station. Proof See Appendix B Proposition 2.2 states that under suitable conditions the distance function is independent of the true distance under the LOS hypothesis and it is therefore theoretically possible to set the false alar rate. Under the conditions stated in the above two propositions, the decision test can be expressed as follows (Appendix C: ˆp r (x log( 2πσˆp r (xdx + ˆσ2 2σ 2 δ, (7 H where ˆσ 2 is the saple variance 2 of the range easureents, that is ˆσ 2 = (r i ˆd 2. Depending on the technique to locate the obile user, the classification of BS s ay not be necessary. Instead soe reliability inforation about the easureents fro each BS ight be required. In this case, the distance value between the LOS and NLOS pdf s can be used as a reliability inforation, which can help us to locate the obile ore accurately. III. SIMULATION RESULTS In this section, the perforance of the non-paraetric NLOS BS identification technique is evaluated. Figure shows the false alar probability, P FA, of the non-paraetric technique for different nubers of easureents ( =3, 5, 0, 5 when the iss detection probability, P MD, is set to The easureent noise is odelled by N (0, 00 2 and the NLOS error is odelled by an exponential rando variable with ean 25. The unit variance Gaussian window is used and the scaling paraeter h is set to 20. In this scenario, we see that we do not need any saples to have a reliable decision. In order to evaluate the perforance of the technique for different NLOS errors, we plot the false alar probability for different NLOS errors. Specifically, we change the ean of the exponential rando variable representing the NLOS error and plot P FA when P MD =0.0 and the nuber of saples is 0. The Gaussian easureent noise and the paraeters of the Parzen window density estiation technique are kept the sae as in the previous case. Figure 2 shows the perforance of the 2 The saple variance is often defined as s 2 = (r i r 2, where r is the saple ean. This definition akes s 2 an unbiased estiate of the population variance. P FA Nuber of Measureents Fig.. False alar probability versus the nuber of easureents when P MD =0.05. P FA Mean of exponential NLOS error (. Fig. 2. False alar probability versus NLOS errors when P MD =0.0 and =0. technique for different NLOS noise levels. Obviously, the test becoes ore successful as the ean (hence the variance of the exponential rando variable increases since it becoes easier to distinguish between the two hypotheses. IV. CONCLUSION A non-paraetric test to deterine whether a given BS is in LOS or NLOS of the MS in question has been proposed. A suitable distance etric between a known easureent error distribution and a non-paraetrically estiated distance distribution has been defined to deterine whether a given BS is within LOS or NLOS of the MS. The perforance of the algorith has been evaluated by siulation experients. Future work includes ore detailed siulation studies under /03/$ IEEE. 2546

4 different NLOS scenarios and coparisons to other techniques [3]. REFERENCES [] J. Caffery, Jr. and G. Stuber, Subscriber location in CDMA cellular networks, IEEE Trans. Vehicular Technology, vol. 47, pp , May 998. [2] J. Caffery, Jr., Wireless location in CDMA cellular radio systes, Kluwer Acadeic Publishers, Boston, [3] J. Borras, P. Hatrack and N. B. Mandaya, Decision theoretic fraework for NLOS identification, Proc. IEEE Vehicular Technology Conference, Spring 998, vol. 2, Ottawa, Canada, pp [4] R. O. Duda, P. E. Hart and D. D. Stork, Pattern classification, Wiley- Interscience, New York, 200. [5] S. Kullback, Inforation theory and statistics, Wiley, New York, 959. [6] C. E. Cook and M. Bernfeld, Radar signal: an introduction to theory and application, Acadeic Press, 970. APPENDIX A. Proof of Proposition 2. Using (3, the KL distance between ˆp r (x and p n (x d can be expressed as ˆp r (x D(ˆp r (x p n (x d = ˆp r (x log dx. (8 p n (x d After inserting (6 and soe anipulations, we get D(ˆp r (x p n (x d = ˆp r (x log( 2πσ ˆp r (x dx + 2σ 2 (x d 2 ˆp r (x dx. (9 When we differentiate (9 with respect to d and equate to zero, we get x ˆpr (x dx ˆd = ˆpr (x dx. (0 Since ˆp r (x is a probability density function, the denoinator is equal to unity. For the nuerator, if we consider a syetric window function, ( x ri x ˆp r (x dx = xφ dx h h = r i ( where we use the fact that window functions integrate to unity. So the optial value of d is the saple ean of the easureents, ˆd = r i, (2 which iniizes (8. B. Proof of Proposition 2.2 The false alar probability is the probability that the KL distance between ˆp r (x and p n (x d exceeds the threshold given that is the true hypothesis. That is, P FA = Pr{D(ˆp r (x p n (x ˆd >δ }. (3 Under, each easureent is equal to su of the true distance and easureent noise, which is a zero ean Gaussian rando variable. In other words, r i = d + n i for i =,...,. Fro Proposition 2., the optial value of d iniizing the KL distance of (4 is given by ˆd = r i. Inserting (2 and (6 in (5, we get D(ˆp r(x p n(x ˆd = ( φ( x ri log h h h φ( x ri log h h φ( x ri dx h dx.(4 ( 2πσ e (x ˆd/(2σ 2 Note that x ˆd = x r i = x d n i, since r i = d+n i under. So defining a new duy variable for the integrals as u = x d, we obtain D(ˆp r(x p n(x ˆd = ( φ( u ni log φ( u ni du h h h h φ( u ( ni log e (u n i /(2σ 2 du, h h 2πσ which solely depends on n,..., n. Therefore, it is possible to set the false alar rate without any inforation about the true distance, d. C. Derivation of Equation (7 The decision test in the Gaussian easureent noise case can be expressed as D(ˆp r (x p n (x ˆd δ, (5 H where ˆd = r i. Siilar to (9, the distance can be expressed as D(ˆp r (x p n (x ˆd = ˆp r (x log( 2πσ ˆp r (x dx + 2σ 2 (x ˆd 2 ˆp r (x dx. (6 The integral in the second ter can be expressed as follows: (x ˆd 2 ˆp r (x dx = (x 2 2 h ˆdx+ ˆd 2 φ( x r i dx, h (7 which can be shown equal to (x ˆd 2 ˆp r (x dx = {h [h 2 h σw+r 2 i 2 ] 2 ˆdr i h + ˆd 2 h }, (8 where σw 2 = x 2 φ(xdx. After cobining the ters we get (x ˆd 2 ˆp r (x dx = h 2 σw 2 +ˆσ 2, ( /03/$ IEEE. 2547

5 where ˆσ 2 = (r i ˆd 2 is the saple variance of the easureents. Inserting (9 in (6, the final decision test can be expressed as in (7 where δ = δ h 2 σ 2 w/(2σ /03/$ IEEE. 2548