Cramer-Rao Bound Analysis of Quantized RSSI Based Localization in Wireless Sensor Networks

Cramer-Rao Bound Analysis of Quantized RSSI Based Localization in Wireless Sensor Networks Hongchi Shi, Xiaoli Li, and Yi Shang Department of Computer Science University of Missouri-Columbia Columbia, MO 6511, USA shih@missouri.edu Dianfu Ma College of Computer Science & Engineering Beihang University Beijing 10008, PRC Abstract Localizing sensor nodes in a distributed system of wireless sensors is an essential process for self-organizing wireless sensor networks. Localization is a fundamental problem in wireless sensor networks, and the behavior of localization has not been thoroughly studied. In this paper, we formulate the quantized received signal strength indicator based localization as a parameter estimation problem and derive the Cramer-Rao lower bound for the localization problem. We study the effect of quantization level and network configuration parameters on the lower bound of localization error variance and understand the relationship between network configuration and localization accuracy. 1. Introduction Distributed systems with hundreds and even thousands of very small, battery-powered, and wirelesslyconnected sensor and actuator nodes are becoming a reality [].Wireless sensor networks are tightly coupled with the physical world, which is an important factor in the tasks they perform [14]. Typical tasks for wireless sensor networks are to send a message to a node at a given location, to retrieve sensor data from nodes in a given region, and to find nodes with sensor data in a given range. Most of these tasks require knowing the positions of the nodes. Sensor network localization has been a fundamental issue of wireless sensor networks and an area of active research in recent years [14]. Localization methods usually follow a three-phase localization model including distance estimation, trilateration or triangulation, and refinement [5]. In the first phase, each sensor node first uses its communication capability to obtain some measurements such as received signal strength indicator (RSSI) to its neighbors to estimate the single-hop distances and then estimates multiple-hop distances to anchor nodes using methods such as a distributed shortest-path distance algorithm. In the second phase, each sensor node uses methods like triangulation to estimate its location using distances to three or more anchor nodes. In the third phase, each sensor node fine-tunes its location according to the constraints on the distances to its neighbors. There are range-free and range-based sensor network localization algorithms [8, 1, 16]. In the range-free approach, the algorithms do not need range hardware support and are immune to range measurement errors while providing less accurate localization results. In the range-based approach, the algorithms require more sophisticated range hardware support while providing more accurate localization results than the range-free algorithms. Much less work has been done on searching and utilizing other range-related information for sensor network localization. In a point-in-triangle localization scheme [4], RSSI values are used for comparing distances. RSSI, measuring the RF energy received, is a type of range-related information and is supported by sensor node hardware such as the MOTE [19]. For the localization purpose, the information provided by RSSI or similar type of measurement is less than range but more than a connectivity-only hop count, and it can be used to improve the accuracy of any range-free localization algorithms [7, 9]. The behavior of localization systems, which can be affected by uncertainties in the network operating environment and the sensor node hardware and software, has not been thoroughly investigated [14]. The uncertainties can usually be statistically modelled [17], which is the basis that we can analyze the localization errors using the estimation theory [1, 11, 18]. Several researchers have analyzed errors of localization in sensor networks using distance information added with simple Gaussian noise [6, 15], and some other researchers have analyzed errors of localization in sensor networks using quantized RSSI values with simple Gaussian noise added [10].

In this paper, we analyze errors of localization in sensor networks using quantized RSSI in a more realistic statistical model. We take into account the effect of the network operating environment on the localization problem in modelling RSSI measurement uncertainties. We also study the effect of quantization level and network setup parameters, such as network size, network density, anchor node placement, and anchor node density, on the localization error variance, which can guide us to set up wireless sensor networks to achieve best localization accuracy.. Quantized RSSI based localization For the localization problem, the network is modelled as a connected, undirected graph G =(V,E), where V is the set of sensor nodes and E is the set of edges connecting neighboring nodes. Each edge e(u, v) is associated with a value z Z (e.g., an RSSI value). Let (x, y) be the unknown coordinates of u V. Let A V be the set of anchors with known coordinates. The localization problem is to find coordinates (x, y) of each node u V \A. The localization problem is solved by setting the edge s z value to either a measured distance, or a unit value (e.g., radius in MDS [16], corrected hop distance in APS [8]) for all the 1-hop (edge) connections. In the quantized RSSI based approach, z is set to p P of quantized RSSI values, where the relationship between P and the set of true distances is an unknown bijective mapping, and it is usually set using an RSSI quantization scheme. In a sensor network of K sensors with locations to be estimated and B anchors with known locations in a plane, sensors can communicate with their neighbors of other sensors and the anchors to obtain RSSI values. Let X = {X kl } be the set of all the RSSI measurements, where X kl is the RSSI value measured by sensor node k of the signal received from sensor node l. The measurements are not perfectly accurate due to effects such as sensing errors and environmental effects [9, 17]. Let each sensor s unknown location be r k =(x k,y k ),k =1,,,K and each anchor s known location be r k =(x k,y k ),k = B +1, B +,, 0. The localization problem can be stated as a problem of estimating the parameters r k,k =1,,,K from the observations (i.e., the measurements X kl s) and the known locations of the anchors (i.e., r k,k = B +1,, 0). Thus, the localization problem can be formulated as a parameter estimation problem.. Cramer-Rao Lower Bound of Localization Error Variance As the localization problem is formulated as a parameter estimation problem, the localization errors can be analyzed through the estimation theory [18]..1. Modelling quantized RSSI errors Sensor nodes in a wireless sensor networks communicate through peer-to-peer links, and pair-wise measurements such as RSSI can be made with these links [9]. The quantized RSSI measurements are subject to the deleterious effects of a fading channel. Received signal strength is attenuated by large scale path losses, frequency selective fading, and shadowing losses [, 9]. Let d kj be the distance between sensor nodes k and j. That is, d kj = (x k x j ) +(y k y j ) (1) The received power at node k transmitted by node j (in dbm) can be formulated as P kj =Π 0 10n p log d kj () where Π 0 (dbm) is the received power at the reference distance. Typically, =1m, and Π 0 is calculated from the free space loss formula [1]. The path loss exponent n p is a function of the environment, typically between and 4. Patwari et al. claim that the power loss has a log-normal distribution [9, 10] with n p a fixed constant. The experimental results used to support their claim are 946 pair-wise RSSI measurements with a DS-SS transmittor and receiver in a network of 44 device locations. In the quantile-quantile plot of comparing the distribution of P kj (Π 0 10n p log d kj ) (the attenuation of the channel) to the Gaussian distribution, they match well in the middle part. At both ends, however, they do not match well. A more realistic assumption is that n p is a Gaussian random variable, i.e., n p N(α, σ) () With this assumption, P kj is a Gaussian random variable P N(Π 0 10α log d kj, [(10 log d kj )σ] ) (4) The conditional probability of having the received power P at device k with location r k transmitted from device j with location r j will be p kj (P )= 1 π(10 log d kj )σ e [P (Π d kj 0 10α log )] 0 [(10 log d kj )σ] (5) Let the measured received power be quantized into S levels with thresholds: + = P 0,P 1,,P S 1,P S =. Let d i be the path length at which the mean received power is P i, i.e., the communication range. The path length d i can be computed from P i as follows: d i = 10 Π 0 P i 10α (6)

with =0and d S =+. The conditional probability of having the received power X i kj in the interval [P i,p i 1 ) will be p(x i kj r k,r j )= Pi 1 P i p kj (P )dp =Φ(D i 1 kj ) Φ(Di kj) (7) where Φ(x) = x 1 π e v dv is the CDF of a univariate zero-mean unit-variance normal distribution and D i kj = α σ ln(d kj /d i ) ln(d kj / ) with Φ(D 0 kj )=1and Φ(DS kj )=0... Cramer-Rao lower bound for localization using quantized RSSI Let R = [r 1,r,,r K ] be the parameter to be estimated and ˆR(X) =[ˆr 1 (X), ˆr (X),, ˆr K (X)] be any unbiased estimate of the parameter from an observation X. Let p(x R) be the probabilistic mapping from the parameter space to the observation space. The elements of the Fisher Information Matrix (FIM) J can be computed as follows: (8) J kl = E( ln p(x R) r k r l ) (9) The lower bound on the variance of minimum-meansquares errors in estimating parameter r k is as follows: σ k = var(ˆr k (X) r k ) J 1 kk (10) The localization problem corresponds to the estimation of sensor node coordinates [R x,r y ] with R x = [x 1,x,,x K ] and R y = [y 1,y,,y K ]. The FIM J for the localization problem can be organized as follows: [ ] Jxx J J = xy Jxy T (11) J yy In localization, nodes k and l make pair-wise observation X kl with probability p(x kl r k,r l ). Let H(k) be the set of nodes that make pair-wise observation with node k. By symmetry, if l H(k), then k H(l). We assume k H(k). Assuming that {X kl } are statistically independent, the logarithm of the joint conditional probability is ln p(x R) = K k= B+1 l H(k),l<k ln p(x kl r k,r l ) (1) Plugging the probabilistic mapping p(x kj r k,r j ) we discussed in Section.1, we can derive the Cramer-Rao lower bound on the localization error variance for localization using quantized RSSI. j H(k) h kj (x k x j) d [J xx ] kl = 4 d kj ln kj (1) (x I H(k) (l)h k x l ) kl k l [J xy ] kl = where d 4 kl ln d kl j H(k) h kj (x k x j)(y k y j) d 4 kj ln d kj I H(k) (l)h kl (x k x l )(y k y l ) d 4 kl ln d kl j H(k) h kj (y k y j) [J yy ] kl = (y I H(k) (l)h k y l ) kl h kj = 1 π and S i=1 [e (D i 1 kj ) I H(k) (l) = 4. Simulation Results d 4 kj ln d kj d 4 kl ln d kl ( α σ Di 1 kj k l k l (14) (15) (D i kj ) ) e ( α σ Di kj )] Φ(D i 1 kj ) Φ(Di kj ) (16) { 1 if l H(k) 0 otherwise (17) Several network parameters affect the Cramer-Rao bound (CRB) of localization error variance. Studying the effects of those parameters can give us insights on setting up sensor networks to achieve best localization accuracy. In this section, we discuss the effects of network size, network density, anchor node placement, anchor node density, and RSSI quantization level on localization error variance using simulation with the CRB we derived in the last section. Let σx k and σy k be the CRB of error variance of estimating x k and y k, respectively. We compute the root-meansquared location error variance of each network, RMS(σ), by RMS(σ) = 1 A (σx A k + σy k ) (18) k=1 The value RMS(σ) gives the average lower-bound variance in errors of estimated sensor node locations. The smaller RMS(σ), the higher confidence we have on the localization result generated by a localization algorithm. To show how the localization error variance distributes in the network, we plot σx k + σy k for a grid network with 5 nodes in Fig. 1(a).

We generate many grid networks with various parameters to understand the effects of network parameters on localization accuracy. To study the effect of a network parameter on localization accuracy, we generate grid networks for various values of that parameter with other parameters fixed, compute an RMS(σ) value for each value of that parameter, and plot a curve of RMS(σ) as that parameter varies. The σ for the path loss exponent n p is set to 0.m in the simulation. 4.1. Effect of RSSI quantization level RSSI quantization level, S, is the parameter used to set thresholds for RSSI value intervals. Fig. 1(b) shows that RMS(σ) decreases as S increases until S reaches a certain value. After that, RMS(σ) cannot be improved by increasing the quantization level. This is due to the noise in the RSSI readings. When the quantized RSSI value interval is comparable to the noise, increasing the quantization level becomes less useful. (a) Localization error variance CRB in a grid network with 5 nodes. 9 8 7 6 5 4 1 0 5 10 15 0 Quantization level on 100-node grid networks with four anchors placed at the four corners of the outermost layer. Figure 1. Simulation result I in a layer so that the number of outer nodes and the number of inner nodes are balanced, RMS(σ) can be kept from increasing too fast as the network size increases. 4.. Effect of network density Network density, dn, is characterized as the number of nodes per square meter. To generate a network of a density, dn, we generate a dn nodes placed in the grid network of area a. We generate 100 networks for each density value and compute the average of RMS(σ) s for those networks as the value of RMS(σ) for that network density. The simulation shows that RMS(σ) decreases when density increases. Referring to the result about the network size effect, to maintain a lower RMS(σ), a large number of nodes are needed to counteract the effect of increasing network size, as shown in Fig. (a)..1.9.8.7.6.5.4...1 1st layer anchors nd layer anchors rd layer anchors 50 70 90 110 10 150 170 190 Network size (a) RMS localization error variance on networks with different sizes when anchors are placed in the 1st, nd, and rd layer (from the outermost toward the center), respectively. The network sizes vary from 50m 50m to 00m 00m. The densities and connectivity degrees are the same except the edge areas..6.5.4...1 1.9 0 10 0 0 40 Anchor density on 100-node grid networks with increasing numbers of anchors randomly selected across the network. The result for each data point is the average RMS computed from 100 networks with the same number of anchor nodes but different positions to neutralize the placement effect. Figure. Simulation result II 4.. Effect of network size We investigate how the localization error variance changes as the network scales in overall size. We evaluate RMS(σ) for a number of networks with a fixed density and a fixed percentage of anchor nodes but with varying sizes. Such networks can be generated by increasing the area the network occupies while increasing the number of sensor nodes so as to keep the node density constant. The results in Fig. (a) shows that RMS(σ) increases when the network size increases, which indicates that network size is a negative factor in localization accuracy. However, moving the anchors from the outermost layer to some inner layer can improve RMS(σ) significantly for larger networks. This suggests that if the anchors are positioned 4.4. Effect of anchor node placement Some experimental localization results suggest that anchors be placed on the perimeter of the network to obtain more accurate localization accuracy [1, 15]. We investigate using CRB how much the placement of anchor nodes affects RMS(σ). We generate networks with a fixed size and density but with different placement schemes: placing anchors in different layers in the network. There are five layers (numbered from the outermost layer toward the center) in a 100-node grid network. Fig. (b) shows that placing four anchors in the center makes the worst RMS(σ), while placing the anchors in the nd or rd layer has the best performance.

4.5. Effect of anchor node density We investigate the effect of percentage of anchor nodes in the network on the localization error variance. The anchor densities are set between 4% and 0%. RMS(σ) decreases significantly as the density of anchors increases before a saturation point, and it remains the same after the saturation point, as shown in Fig. (b).. 1.8 1.6 1.4 1. 0.5 1 1.5 Network density (a) RMS localization error variance on 100m 100m networks with increasing numbers of nodes. The four anchors are placed on the corners of the outermost layer. 5. Conclusions 4.5 4.5.5 1 4 5 Anchor placement layer on 100-node grid networks with four anchors placed at the four corners of layers numbered from the outermost toward the center. Figure. Simulation result III Localization is a fundamental problem of deploying wireless sensor networks for many applications. Although many algorithms have been developed to solve the localization problem, the fundamental characterization of localization error behaviors still needs to be studied. In this paper, we have analyzed the variance of localization errors using Cramer-Rao lower bound and have studied the effects of network parameters on localization accuracy, which provides us insights on how to set the controllable parameters of a sensor network for the best possible localization accuracy. Acknowledgements The work presented was partially supported by the National Science Foundation under grant CNS-0486. References [1] H. Cramer. Mathematical Methods of Statistics. Princeton University Press, Princeton, NJ, 1946. [] D. Ganesan, B. Krishnamachari, A. Woo, D. Culler, D. Estrin, and S. Wicker. An empirical study of epidemic algorithms in large scale multihop wireless networks. Technical report, UCLA Computer Science Department, 00. ucla/csd-tr-0-001. [] H. Hashemi. The indoor radio propagation channel. Proceedings of the IEEE, 81(7):94 968, 199. [4] T. He, C. Huang, B. Blum, J. Stankovic, and T. Abdelzaher. Range-free localization schemes in large scale sensor networks. In Proceedings of ACM/IEEE International Conference on Mobile Computing and Networking (MobiCon), July 00. [5] K. Langendoen and N. Reijers. Distributed localization in wireless sensor networks: A quantitative comparison. Computer Networks, 00. [6] E. G. Larsson. Cramer-Rao bound analysis of distributed positioning in sensor networks. IEEE Signal Processing Letters, 11():4 7, 004. [7] X. Li, H. Shi, and Y. Shang. Partial-range-aware localization algorithm for ad-hoc wireless sensor networks. In Proceedings of the 9th Annual IEEE Conference on Local Computer Networks (LCN 004), Tampa, FL, 004. [8] D. Niculescu and B. Nath. Ad-hoc positioning system. In Proceedings of GlobeCom 001, November 001. [9] N. Patwari and A. O. Hero. Using proximity and quantized RSS for sensor localization in wirelss networks. In Proceedings of the Second ACM International Workshop on Wireless Sensor Networks and Applications, San Diego, CA, 00. [10] N. Patwari, A. O. Hero, M. Perkins, N. S. Correal, and R. J. O Dea. Relative location estimation in wireless sensor networks. IEEE Transactions on Signal Processing, 51(8):17 148, 00. [11] C. R. Rao. Information and accuracy attainable in the estimation of statistical parameters. Bulletin of the Calcutta Mathematical Society, 7:81 91, 1945. [1] T. Rappaport. Wireless Communications: Principles and Practice. Prentice-Hall, 00. [1] C. Savarese, J. Rabay, and K. Langendoen. Robust positioning algorithms for distributed ad-hoc wireless sensor networks. In Proceedings of 00 USENIX Technical Annual Conference, June 00. [14] A. Savvides, M. Srivastava, L. Girod, and D. Estrin. Localization in sensor networks. In Wireless Sensor Networks. Wiley/IEEE Press, 004. [15] A. Savvides, W.Garber, S. Adlakha, R. Moses, and M. B. Srivastava. On the error characteristics of mulithop node localization in ad-hoc sensor networks. In Proceedings of Information Processing in Sensor Networks (IPSN0), Palo Alto, CA, April 00. [16] Y. Shang, W. Ruml, Y. Zhang, and M. Fromherz. Localization from mere connectivity. In Proceedings of Fourth ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc), Annapolis, MD, 00. [17] S. Slijepcevic, S. Megerian, and M. Potkonjak. Location errors in wireless embedded sensor networks: Sources, models, and effects on applications. Mobile Computing and Communications Review, 6():67 78, 00. [18] H. L. van Trees. Detection, Estimation, and Modulation Theory: Part I. Wiley, New York, NY, 1968. [19] K. Whitehouse and D. Culler. Calibration as parameter estimation in sensor networks. In Proceedings of the 00 ACM International Workshop on Wireless Sensor Network and Applications, Atlanta, GA, September 00.