STRATEGIC WHITE PAPER GEOLOCATION BOUNDS FOR RECEIVED SIGNAL STRENGTH (RSS) IN CORRELATED SHADOW FADING Laurence Mailaender, LGS Innovations LLC, Florham Park, NJ Abstract Cramer-Rao bounds for geolocation based on Received Signal Strength have previously been studied under a Log Normal fading model. This paper develops new bounds under several different assumptions to identify the most promising scenarios. We introduce a spatially Correlated Log Normal (CLN) model and find the related bound. We derive a new bound for a Relative Received Signal Strength (RRSS) technique based on power ratios. Finally, for moving receivers, we model data collection over a time window where multiple independent fading instantiations are observed. Results show that RSS location accuracy improves significantly (25%) in realistic fading correlation, and more under stronger correlation. We prove that RRSS geolocation is no better than RSS in CLN fading. For moving users, processing a time window of 20 samples reduced location error by up to a factor of 4.
TABLE OF CONTENTS I. INTRODUCTION 01 II. RSS IN LN FADING 02 III. RSS IN CLN FADING 04 IV. RRSS IN CLN FADING 05 V. RELATION BETWEEN J CLN AND J RRSS-CLN 06 VI. MOVING USER WITH TEMPORAL COLLECTION 07 VII. NUMERICAL RESULTS 09 VIII. CONCLUSIONS 10 IX. REFERENCES 11 ii
I. INTRODUCTION Wireless geolocation, or position determination may be accomplished via many techniques, such as Time- Difference-of-Arrival (TDoA), Angle-of-Arrival (AOA), or Received Signal Strength (RSS). While ToA and TDoA are generally considered the most accurate, RSS techniques are interesting for their simplicity and generality. Note that RSS positioning can be accomplished from a set of reference stations (at known positions) radiating any set of waveforms (synchronization is not required), as long as the transmit power and propagation law are known. This motivates the idea of performing RSS-type positioning from a set of Signals-of-Opportunity (SoO) that happen to be in a local area (e.g. commercial radio, TV sources, etc.). The large number of potential sources will increase the location accuracy. The Cramer-Rao Lower Bounds (CRLB) for ranging [1] and positioning [2] error for RSS systems have previously been found, assuming a Log Normal (LN) model for fading. These results show that the RSS ranging error is linearly proportional to the distance between the user and a reference station; the accuracy is poor relative to TDoA under typical conditions. This paper derives CRLBs corresponding to several new assumptions. First, we introduce a Correlated Log Normal (CLN) model which acknowledges that the local environment around the mobile user induces some spatial correlation in the shadow fading to distant bases/reference stations. Then, we provide a new CRLB for a relative RSS (RRSS) system under the CLN model. We note that [3] also includes correlated log normal variables in a CRLB, but this models correlation between forward and reverse links, which is different than the spatial correlation model used here. Several authors have previously proposed location algorithms based on RRSS. Specifically, [4] develops a power ratio approach, but log normal fading is not included, and [5] gives an algorithm for Suzuki fading. Neither [4,5] gives a CRLB as we do here. In [6] the author develops the CRLB for LN in the important case of unknown pathloss exponent, and examines a LSE estimator which is biased and actually beats the CRLB (which applies to unbiased estimators). The bias in range estimation under LN conditions has also been noted in [2,7]. Finally, if we assume the user is moving, we can collect data over a time window incorporating uncorrelated measurements of each SoO source. After this research was completed, we became aware of some recent work [8,9] exploiting spatial correlation. In [8] spatial correlation is assumed to exponentially decaying with physical distance, and an ad hoc network is modeled. In [9], an explicit correlation coefficient is used as in our approach, but the results concentrate on least-squares estimation. 1
II. RSS IN LN FADING In this section we review the CRLB derivations for ranging and location under the Log Normal (LN) model. Assume the power received from the k-th source is, (2.1) P k = a k d k -y l k where a k is the link budget constant (e.g. transmit power, antenna gain, etc., all assumed known) for the k-th user, d k is the distance, is the propagation law path loss exponent (assumed known), and l k is a Log Normal r.v. where l k = 10 1nk = and n k is zero-mean real Gaussian with E{n n2 } = σ 2, E{n n } = 0,k j. The LN model (2.1) implicitly assumes sufficient SNR and shad k j averaging time to render thermal noise and fast fading insignificant. Consider the transformation, (2.2) Hence, r k is conditionally Gaussian with log-likelihood, (2.3) Taking the derivative w.r.t. d k (2.4) It is straightforward to show from (2.2) that (2.5) where (10). The Fisher matrix is formed from the expectation (2.6) The ranging bound is, (2.7) which is linearly proportional to distance, d. This expression agrees exactly with [1]. Next consider a set of independent power measurements of K sensors, having Log Likelihood, (2.8) where. Here (x, y) is the unknown position to be estimated, and (x k, y k ) is the position of the k-th sensor. Taking the derivative w.r.t. x, (2.9) 2
Then, with we find, (2.10) Then which agrees with the result in [2] 3
III. RSS IN CLN FADING We now introduce a Correlated Log Normal (CLN) model where the Gaussian r.v. is broken into two parts [9,10], (3.1) Here n ~,n ~ k are i.i.d real, zero-mean Gaussian with variance σ 2,and E{n n } = shad i j ρσ2, i j. Note that the r.v. shad n~ contributes to all K measurements. The motivation for this model is that the local environment is homogenous (say, inside a particular building) and contributes the same attenuation to all links, however, farther away the environment is dissimilar and the fading differs per link. The Log-Likelihood for the received measurements (2.2) is now, (3.2) (3.3) Taking the derivative w.r.t,θ, (3.4) (3.5) (3.6) When ρ = 0 we see that equations (3.6) and (2.10) are equal. 4
IV. RRSS IN CLN FADING If the receiver has not been calibrated (2.1) becomes (4.1) where c is an unknown constant. In RRSS, assume we take one of the measurements as a reference, and form K-1 power ratios as the basis for positioning. Here, we take sensor 1 as the reference. (4.2) for k = {2,---,K}. Note that c and the r.v. n ~ cancel out in this expression. Define. Then, and the covariance is, (4.3) Proceeding as in (2.2), let (4.4) Define r = [r 2,---,r K ] T which is Gaussian distributed as ( 1 (,d),r Z. Returning to (3.2) with R Z as the covariance, (4.5) Consider the k-th entry of the first term, (4.6) where f x is the derivative of f w.r.t. x. Repeating this for y we find, (4.7) where, (4.8) and finally, (4.9) 5
V. RELATION BETWEEN JCLN AND JRRSS-CLN Define H = [h 1,H 2 ] and note that using the matrix inversion lemma we can write (5.1) Then expanding (3.6), we find, (5.2) Turning to (4.9), we can write, (5.3) Expanding terms we find, (5.4) Thus equations (5.2) and (5.4) would be equal if, (5.5) Hence, equality occurs (sufficient condition) when ρ =1, otherwise the l.h.s. will be smaller, giving the result, (5.6) Equality also occurs under the following source symmetry condition. Note that (5.2) and (5.4) are of the form, (5.7) only differing in the constant, c. However, if H is such that (5.8) then equality occurs for any value of c (i.e. any value of ρ.) The symmetry condition on l.h.s. of (5.8) is fulfilled (sufficient condition) when the vector sum of all the source position vectors is zero (r.h.s.). In the special case of sources on a grid centered at the origin it is intuitively obvious that the symmetry condition is fulfilled. 6
VI. MOVING USER WITH TEMPORAL COLLECTION Assume the user collects data while moving distance D along a straight line, and the shadow fading is uncorrelated after d I meters. M = 1 + [D /d I ] uncorrelated temporal measurements of each source are made; the CLN model defines the spatial correlation at each position. We consider two separate cases, denoted centroid and inertial. The M measurements are made at locations (x m, y m ) which are approximately equal to the centroid of the positions, denoted (x c, y c ). Write the total position error as, (6.1) where the first term is bounded by the CRLB for estimation of the centroid position, and the second term is a deterministic bias. Let the mth measurement of the K SoO under CLN be, (6.2) where d c is the vector of distances from the centroid to the SoO positions, E { n m n mt } is given by (3.3), and E { n m n mt } = 0, m m'. Taking ϴ = [x c,y c ] T, the likelihood for the total measurement interval is, (6.3) Therefore the centroid case is like (3.6) but with an M-fold SNR improvement, (6.4) The total position error is plus the second term of (6.1). Next, we assume that the moving user has an inertial sensor, and perfectly knows the differences in position among the M measurement locations. The m-th measurement is taken at position, (6.5) where, are known from the sensor, and we will estimate the unknown final position,. The m-th received vector is, (6.6) where the distance is a function of, x M, y M, (6.7) The log-likelihood for the entire observation is, (6.8) 7
where A B denotes a Kronecker product. Taking the derivative, (6.9) and the Fisher matrix for the inertial case is, (6.10) 8
VII. NUMERICAL RESULTS In our numerical results, the user is assumed to lie at the center of a rectangular grid of SoO ranging sources. The radio propagation is characterized by pathloss exponent = 4, and shadow fading standard deviation σ shad = 8 (db) unless otherwise noted. The rectangular grid is in principle infinite, but here extends ±6 km along x and y axes. Note that the number of sources varies as the grid spacing changes (enough sources to fill the 144 km 2 ). Figure 1 shows the CRLB of (2.10) in meters as a function of the grid spacing. Error is linearly proportional with the shadowing: 8 db shadowing has twice the error of 4 db. To achieve errors on the order of 100m we will need to have SoO placed every 0.5-1.0 km. Figure 1 also lets us say something about the scaling of location error with increasing number of sources. Reducing the SoO distance from 2 to 1 km corresponds to a 4- fold increase in the number of SoO; note that the location error is approximately halved. This suggests the error improves approximately as the square root of the number of sources. See also [11]. Figure 1: CRLB vs Grid Spacing Figure 2: CRLB for RSS and RRSS under CLN Model In Figure 2, we plot the error for RSS (3.6) and RRSS (4.9) under the CLN model with sources on a grid. Location error decreases as the spatial correlation increases. Four curves are plotted, but the RSS and RRSS results are equal for sources on a grid, as expected from the sufficient condition in (5.8). Assuming ρ= 0.5, the error at 2 km spacing decreases from about 440m (ρ = 0.0 ) to 325m, a 25% gain from modeling the correlation. Performance gains increase noticeably for even stronger correlations. Figure 3 considers a user moving in a straight line, making a new uncorrelated measurement every d I = 15m, with spatial correlation=0.5. The x-axis shows the total number of measurements included, with grid spacing of 0.5 through 2 km. The location error improves dramatically when including up to 10 measurements, and levels off after about 30. When the error is about 25m or less, the inertial method begins to show very nice gains over the centroid method. Comparing the location error for 1 vs. 20 measurements, we find the error is reduced to about ¼ of the original level. Figure 3: CRLB with Temporal Collection 9
VIII. CONCLUSIONS This paper considered the CRLB when there is correlated shadow fading between the multiple received links. Location error decreases about 25% when typical correlation is introduced. We developed a new CRLB for the RRSS case and proved it is no lower than the RSS case; the only advantage of RRSS is that it can be used with an uncalibrated receiver. Incorporating measurements over time when the user is moving is the most promising technique identified in this paper. This can reduce the error by a factor of 4, bringing the total error below 20m using SoO spaced every 0.5 km. ABOUT LGS INNOVATIONS LGS Innovations LLC solves the most complex networking and communications challenges facing the U.S. Federal Government. Building on its Bell Labs heritage, LGS Innovations delivers groundbreaking research and advanced networking and communications solutions that provide an information advantage and contribute to the mission success of its customers. Solutions include Infrastructure & Installation; Video Teleconferencing and Surveillance; Tactical Communications; Wireless/Mobility; 4G/LTE; Enterprise, Optical and Data Networking; and research and development in Advanced Multimedia/RF, Cybersecurity, sensing technologies, and Photonics. An independent subsidiary of Alcatel-Lucent dedicated solely to serving the U.S. Federal Government, LGS Innovations is headquartered in Herndon, Virginia, with offices in Colorado,Illinois, Maryland, New Jersey, and North Carolina. To learn more about LGS Innovations, visit www.lgsinnovations.com. LGS Innovations: The Network Experts 10
IX. REFERENCES 1 Y. Qi and H. Kobayashi, On relation among time delay and signal strength geolocation methods, Globecom 2003. 2 N. Patwari, A. O. Hero, et. al., Relative location estimation in wireless sensor networks, IEEE Trans on Signal processing, Aug 2003. 3 N. Patwari and A. O. Hero, Signal strength localization bounds in ad hoc and sensor networks when transmit powers are random, IEEE Workshop on Sensor Array, 2006. 4 H. Saarnisaari and T. Braysy, Systematic errors and location accuracy in wireless networks, EURASIP Journal Applied Sig. Proc., 2006. 5 S. Namik, U. Ferner, and K. W. Sowerby, Localization in harsh propagation environments, AusCTW 2008. 6 X. Li, RSS-based location estimation with unknown pathloss model, IEEE Trans on Wireless Comm, Dec 2006. 7 S. D. Chitte, S. Dasgupta, and Z. Ding, Distance estimation from received signal strength under log-normal shadowing: Bias and variance, IEEE Signal Processing Letters, March 2009. 8 N. Patwari and P. Agrawal, Effects of correlated shadowing: Connectivity, localization, and RF tomography, Int l Conf. on Info. Processing and Sensor Networks, 2008. 9 J. H. Lee and R. M. Buehrer, Location estimation using differential RSS with spatially correlated shadowing, Globecomm, 2009. 10 A. J. Viterbi, CDMA: Principles of Spread Spectrum Communications, Addison-Wesley, 1995. 11 L. Mailaender, On the CRLB scaling law for received signal strength (RSS) geolocation, to appear in CISS 2011. 2012 LGS INNOVATIONS LLC - ALL RIGHTS RESERVED LGS, LGS INNOVATIONS, AND THE LGS INNOVATIONS LOGO ARE TRADEMARKS OF LGS INNOVATIONS LLC.