On the A Priori Localization Errors in Sensor Networks P. Uday Kiran, S. Sundhar Ram and D. Manjunath Department of Electrical Engineering Indian Institute of Technology, Bombay Mumbai INDIA 400 076 udayk,sram,[email protected]

Abstract— We perform a priori localization error analysis for unbiased estimators for different node and target localization schemes for an ad-hoc network deployment model. We derive the a priori Cramer-Rao lower bound and express it as a function of network parameters. The analysis performed is distinct from earlier studies, where fixed and unknown sources are localized in known networks. We also compare the a priori error of some simple estimators (not necesarily unbiased) against the a priori lower bounds that we derive in this paper. The latter is to obtain a measure of the effectiveness of the known localization estimators. We also investigate the similarities in node and source localization problems through an extensive literature survey. An important aspect of our analysis is that the measurement errors could be proportional to the distance of the sensor from the target or node.

I. I NTRODUCTION Networked sensing offers efficient solutions compared to traditional sensing in terms of reliability, reachability, and adaptability. Advances in wireless technology and microfabrication are facilitating the development of small-sized, low-cost sensor nodes that can operate at low power and communicate untethered even in hostile environments. A typical wireless sensor network (WSN) consists of a large number of sensor nodes capable of communicating with each other over a wireless medium. Each sensor node is equipped with on-board sensing, processing, storage and communication capabilities. Nodes in a sensor network can be one of three types—sensor nodes that do the actual sensing and communicate it to anchor nodes that may process and fuse the data locally, which in turn is communicated to the gateway nodes. Most sensor network applications involve obtaining local samples about a global phenomenon in the operational area of the network, possibly processing the observed samples to derive information and then communicating this information to the external world. Of course, all this should be done efficiently. See [1] for a comprehensive list of possible applications of sensor networks. Target, or source, localization is a canonical sensor network function [2]. Here, the sensor nodes obtain some information, possibly with error, about the location of the target that is in the sensing field. An example of such information could be the distance of the target from the sensor, also called the range information. This information is then fused to ‘localize’ the target (or source). Such fusion algorithms will require that

each of the sensor nodes know its location, possibly with error. This implies that in a randomly deployed network the sensor nodes need to first ‘localize’ themselves. In sensor network literature, this is called ‘self localization’. Self localization is achieved by nodes making measurements to neighbors who know their locations, possibly with error and then fusing these measurements. Thus we see that localization, target or self, is a canonical problem of sensor networks in which we have some entities that are aware of their locations and some entities whose locations need to be estimated by noisy measurements. We will investigate the relation between target and self localizations in Section II. We will then develop the analysis of localizations in random sensor networks using the target localization framework. The analogy to self localization will be obvious. In this paper, our focus is in the a priori characterization of the localization error in a randomly deployed network. We analyze the following four different models of localization systems which differ in their data collection and measurement schemes. 1) Localization systems that use range measurements from the closest sensors to the target. 2) Localization systems that use range measurements from  of the target. all the sensors within a distance 3) Localization systems that use the range-difference measurements from the closest sensors to the target. 4) Localization systems that use the range-difference mea  surements from all the sensors within a distance of of the target. We obtain the a priori lower bound on the localization error for any unbiased localization estimator as a function of network parameters like deployment density and measurement errors. This is distinct from obtaining bounds in a given network where the node locations are known and we are localizing a fixed but unknown target. Much of the literature on localization essentially deal with such fixed setups and suggest different localization algorithms. See, for example, [3], [4]. In contrast, the a priori analysis that we present in this paper may be viewed as a ‘pre-deployment’ analysis and is expected to help in the deployment process, e.g., in selecting the node density. To understand the difference in the analysis method when the sensor locations are known and when we know only the

distribution of the sensor locations, consider a sensor network that performs target localization using range measurements to the target. Clearly, the estimation error depends on the location of the sensor nodes but in an a priori analysis, these are not known. Instead, we know the distribution of the node locations. Thus the actual value of the ranges and the sensor locations are random variables which have a nonlinear relationship. The parameters of this nonlinear relation are the target coordinates. Thus the localization can be viewed as estimating the relationship between the ranges and the sensor coordinates from noisy observation of the ranges. Such a problem was first articulated in the statistics literature. See [5] for a discussion on a generalized version of such a problem. The relation between the variables is called a structural relation and the analysis is a structural analysis. In contrast, in a fixed system, the actual value of the observed quantity and the sensor locations are fixed and are related through the parameters. Such a relation is called a functional relation and the analysis is a functional analysis. In this paper we perform a structural analysis of target localization in a randomly deployed sensor network. To the best of our knowledge very few similar studies have been carried out. [6] examines the a priori error in localization in a one dimensional network when the localization is performed using a specific iterative estimator that provides the set of all possible locations for the unknown nodes from connectivity constraints of the deployed network. The error in this estimator is defined as the total length of the solution set and this error is then analyzed a priori by assuming that the sensors are distributed as a Poisson process along the real line. [7] simulated a more general deployment model in which sensors are uniformly distributed in a region and the number of sensors in the region is proportional to its area. A number of instances of such deployments are simulated, the CramerRao lower bound on the estimation error is obtained for each of instance, and finally these lower bounds are then combined to obtain the a priori error. Here only simulation results are presented, closed form expressions are not provided. A further contribution in this paper is the measurement error model. While almost all other analyses assume that all measurement errors are i.i.d. Gaussian with zero mean, we allow the error to be proportional to the distance between the sensor and the target (or node). We believe that this is a more realistic assumption for many classes of measurements. The rest of the paper is organized as follows. In Section II we draw comparisons between target and node localization problems and define a common platform to analyze both. We then review the literature for these problems. In Section III we provide a mathematical framework for our problem and define models for the network parameters. In Section IV and in Section V we evaluate the a priori Cramer-Rao lower bound (CRLB) for the localization systems based on range measurements and range-difference measurements. The efficiency of some simple estimators are also studied in these sections. In the final section (Section VI) we present our conclusions.

II. TARGET AND N ODE L OCALIZATION : A N I NTEGRATED V IEW A typical target localization system consists of a set of sensor nodes which are aware of their own location, perhaps with some error. These nodes make distance or angle measurements to one or more targets and these measurements are then fused to localize the target. In node localization, we obtain the locations of the set of ‘blindfolded’ [8] nodes with the help of measurements to the neighbors and to the reference (anchor) nodes which know their location. Though target and node localization define the same problem, they have their own unique issues and specifications. A basic difference is that in target localization the targets may not be cooperative and do not participate in the localization. Thus it is upto the sensor nodes to make measurements and fuse them to estimate the target location. In contrast, in node localization the target entities (the blindfolded nodes) are co-operative and actively participate in the localization procedure. This has allowed the development of iterative node localization algorithms that can produce fairly accurate estimates. Another important difference is the node density. In the case of target localization the sensor node density is generally high and there is always a high probability that there are a sufficient number of nodes localizing a target. This is not generally true in node localization. Generally, the anchor node density is much lower than the node density [9]. A third distinction between node and target localization is that the node localization is performed only once after the network is deployed and hence the computational complexity is not a major issue. While for target localization, computational complexity is clearly a major issue because localization has to be performed throughout the lifetime of the network. To aid in developing an integrated view for target and node localization we review the existing literature on node localization. It must be noted that the references mentioned below perform functional analysis and suggest methods to self localize deployed networks. Three different classifications for localization algorithms exist in literature: 1) Coarse and fine grained localization algorithms [9]: Coarse grained algorithms are those that use only connectivity information to provide coarse estimates for the node locations [10], [9], [6], [11]. Fine grained algorithms use angle and/or range measurements to estimate the node locations [12], [13], [8], [14], [15]. [16] provides a comparison between the CRLB of the two classes of algorithms by considering proximity based techniques as a quantized version of the ranging technique. [13] provides a novel method for localization without using beacons but by using angle-ofarrival (AOA) and time-of-arrival (TOA) measurements to nodes whose locations may or may not be known. 2) Centralized and distributed algorithms: Centralized algorithms [10], [11], [8] are those that run at a single node in the network which fuses the data collected from the nodes to obtain the node locations. In distributed

algorithms [17], [15], [12], [18], [19] the sensors obtain their locations in a distributed cooperative manner. [20] provides a quantitative comparison between the distributed algorithms of [12], [18] and [19]. Clearly, distributed algorithms are preferred over centralized algorithms because of scalability. 3) Iterative and non-iterative algorithms: Almost all the algorithms suggested are iterative and involve repeated refining of the node location estimates [12], [11], [8], [15]. [9] is a non-iterative coarse grained technique. Most fine grained node localization algorithms can be viewed as a a sequence of phases [20]. In the first phase angle and/or distance measures are made between the ‘blindfolded’ nodes and the anchor nodes. In the second phase an initial node location is obtained. This is refined in the third phase using new information obtained from neighbors which were earlier blindfolded. This last phase may be an iterative process. Thus node localization algorithms involve repeated target localization like operations and we can state the generic localization problem as follows. Given the locations of some nodes with known locations and a target or target that activates these nodes, estimate the location of the target. In this paper we analyze this problem and we will refer to it as a target localization problem. Typically, the sensor nodes obtain information about the target in one or more of the following forms: angle-of-arrival (AOA) measurements, time-of-flight (TOF) measurements, time-difference-of-arrival (TDOA) measurements or receivedsignal-strength (RSS) measurements. Since the type and extent of information available about the target location in each measurement class is different, separate localization algorithms have been proposed for each of these measurements. TOF and RSS translate into range measurements while TDOA translates into a ‘range difference’ measurement. TDOA Measurements: Traditional sensor networks were based on passive sensors and hence only TDOA measurements were available. [21], [22] discuss methods of estimating the TDOA from the received signal. Further, [23] showed that the error in TDOA measurements can be modeled to be independent of the range difference measurements. If the propagation speed is known, the range-difference, the difference in the distance between target and two sensors, can be obtained from the TDOA. A large body of literature exists on the estimation of target location from the range differences. The locus of a point which has a fixed range-difference from two points is a hyperboloid in 3-dimensional space and hyperbola in 2dimensional space. Both maximum likelihood estimators and least square estimators have been proposed to locate the target from TDOA measurements. Least square estimation can be viewed as finding the point that is closest to a set of hyperboloids. This is a nontrivial minimization problem and various techniques have been suggested to find approximate solutions. Spherical intersection method [24], spherical interpolation [3], quadratic correction least square approach [25] and a linear correction least square approach [4] obtain an approximate solution by minimizing

a spherical least square error function. [26] considers sets of three sensors and locates the target as the intersection of planes. This method is referred as the plane intersection method. (We will compare the performance of this estimator against the a priori Cramer-Rao lower bound for an unbiased estimator that we obtain in this paper.) [27] obtains a location estimate by minimizing a linear error function for a fixed sensor geometry. [28], [29], [30] and [31] propose maximum likelihood based solutions. While [28], [29] are closed form solutions, [30] and [31] are iterative techniques. TOF and RSS Measurements: In sensor networks with passive sensors, it is not possible to measure the TOF unless the emission time is known. However, with the development of active sensors TOF ranging has been successfully implemented using acoustics, ultrasound or RF signals [32], [33]. [34] describes a sensor network implementation with TOF and RSS measurements while [35] performs simulation experiments with RSS measurements. Much of the node localization algorithms in the literature are based on ranging using RSS and TOF. For this case, a least square estimator (LSE) for the target can be easily derived ( [35]). If the characteristics of the measurement error are known, a weighted LSE that performs better than the LSE can also be defined. In Section IV-C we compare the performance of the the weighted LSE and the a priori CRLB that we derive in this paper. AOA Measurements: In the far field only the AOA is measurable. Extensive literature exists on the estimation of AOA from the received signal. A review of different techniques is provided in [36]. A number of techniques exist for estimating the target from the AOA measurements. See [37] for a list of references. Other Measurements: Apart from these main classes, [37] has developed a one step procedure to localize directly from the observed signal while [13] uses a combination of TOA and DOA measurements to localize the target. III. P ROBLEM D ESCRIPTION

AND

P RELIMINARIES

In this section we describe the problem formulation and the analysis method for localization on a two dimensional plane. The analysis can be easily extended to the three dimensional case. We first describe the notation. Uppercase is used to denote random variables and lower case is used for parameters and non random variables.  will denote the coordinates of the target location and   will denote its estimate. The  location the sensor node  will be   and we define  ,   as the oflength and angle respectively of the vector joining the     target and sensor node  . We will denote by     "! the range-difference to target between sensor nodes  and # . $  will denote the range measurement to target as made by sensor node  and $   will denote the range-difference as measured by sensor nodes  and # . The error in these measurements are %  and %   respectively. Index  for a sensor node refers to

021 D -, 576 D 9 0214B=576"BC9 . ./ B ; 0214A56@A:9 () ; A & 9 '& 021 ? 576 ? '

;=D ;@?

sensor nodes

_      ^^=^     n     N T 7oqp Y p k"r ]s oqt Y t k"r ] R Y  [ \ X T T a a (4) H Given that there are sensors in a circle of radius u around the target, the joint density of the Cartesian n   P  c S T  (5) !  F USwvl   !  ^= U^S^ h   S . In this case, the joint density of I   I   I  _ I   I    ^=^^ I   a  a I S I S S ^^=^ I T T T a P S T a a S S  T ` T where I8Wj and fxh yj P. _          ^=^^     T a a S S   where    F are such that  

021<=5768<>9 ;3

Illustrating the notations.

the FEG -closest sensor to the target. Figure 1 illustrates this notation. We consider localization in networks under different measurement error and sensor (or node) activation models. These are described below. A. Random Deployment of the Sensor Nodes We assume that the sensor node locations form a spatial Poisson process in two dimensions. Note that [6] considers a Poisson distribution of the nodes on a straight line while [7] approximates such a distribution via simulations. The Poisson distribution assumption gives us the following.

H

The joint density of the locations of the closest to the target is given by

coordinates of the sensor locations is given by

0214* 3*+57683:9

Fig. 1.

H

The probability that there are sensor nodes in a circle of radius I around the target, JLKM , is

J K   O NQPRWVI"S" UT XZYL[8\ K]"^ (1) H The density of the range   and angle   are given by _ I   ` N  P  I S  YRa X YL[8\ Kb]   !dc V _   O  ` c ^ (2) P  `P. where I8Wegf and fih Wj H The joint density of the ranges and angles relative to the     ^=^=^      given by target, T T _ I   a I a    ^=^^ I    N T I I ^=^^ I Kk] (3) T T a a S S a S  T XZYL`[8\ ^^=^ jmI and flj  e P . Note where fljmI jmI j S T   a that given   , the location of the EOG sensor node is completely described by I8 F .

T

B. Measurement Error Models In target localization, the signals from the target are measured by the sensor nodes. These measurements are processed to obtain range and range-difference measurements. Owing to factors like channel conditions, measurement limitations of the sensor, etc., the measurements will have an error associated with them. There are many measurement error models available in the literature. We discuss these below and suggest suitable generalizations. Our analysis will be for these generalizations. 1) For Range Measurements: [34] models the error in range measurements obtained from TOF as independent Gaussian with zero mean and error in the range measurements obtained from RSS as a log normal from indoor office area measurement experiments. [35] models the error in range measurements obtained from RSS as an Gaussian with variance dependent on I  . We generalize to model the range measurement at node  as Gaussian distributed with mean I  , and the standard deviation proportional to the actual distance between the target and the node i.e., z i|{ }~vmI  . We believe that this model is reasonable because sensor nodes further from the target are prone to higher measurement errors. The constant { } represents the residual error that is independent of the range and captures errors in the knowledge of the location of the sensors. We thus have,

 €ƒ‚ I   { S O}„v…I  S ^

2) For Range-Difference Measurements: When the sensor nodes obtain range-difference measurements, we assume that the range-differences are made with respect to the closest (first) sensor. The model that we consider has been used in [4] and is an adaptation of the model from [23]. Here, the joint distribution of the range-difference measurements with covariance matrix † , is given by

_     ^^=^    n     ^^=^          S a T a a a “ T T “ ]M‘’ KU’ ” •w–˜—  ‘7’ Y KU’ ”  ‡Uˆ ` T‹Šwc Œ: P‰ †x a Ž S XZY Y ^=^^    œž and I ™ ›š I   ^=^=^ I  œ2 where  ™ ›š    S T S a T a a refer to this model as joint a Gaussian error model.

. We

C. Sensor Activation Clearly, not all nodes in the network will be involved in a localization. We consider two models for selecting the active sensor nodes that will participate in the localization. [35] refers to sensor activation models as ‘data collection techniques’.

1) Closest ‚ activation model (C ‚ AM) : We assume the nodes closest to the target are used. Remark 1: The closest sensors can be identified by using a priori estimates of the target location if available, like in target tracking applications. In other applications where no a priori information about the target location is available, it is not possible to identify the closest sensors for activation using just noisy range measurements. One might suggest that a fairly simple method would be to use the lowest range measurements. However, in Appendix A, we obtain results to argue that the probability that -closest will not produce the lowest range measurements is significant. The utility of analyzing this model is in the fact that with range dependent error, the lower bound obtained on the estimation error will be lower than that obtained if any other sensors are used, however they may be chosen. We will also show that for the case of range-differences, the lower bound on the error will be independent of the sensors used in the localization. 2) Fixed radius activation model (FRAM) : In this model   from only those sensors that are within a radius the target are assumed to be activated and used in the estimation. In our analysis we assume that the data from the that give range measurements of greater than u sensors are not discarded.

D. Cramer-Rao Lower Bound We obtain the lower bounds on the a priori error estimates as the Cramer-Rao lower bounds (CRLB). Under general conditions the MLE asymptotically achieves the bound and the CRLB is a good approximation for the variance of the MLE ( [38]). Thus the a priori CRLB calculated in this paper can help in the deployment design of sensor networks.     is the In our localization problem, the target location _ parameter vector that is to be estimated. Let Ÿ denote the joint density function of the measurements. The CRLB is given by

 ¡£¢¥¤8¦§©ª¬¨ « ­w®¯ §¥° ¨ ª¬¨ «

¸ ­Q®¯ §uª ¨ °±¨ « ²³µ´·¶ ¢¥¤8¦§°©¨ « ¹»º8¼ º"¹»½¼ ½¿¾l¹»º8¼ ½@¹»½=¼ ºÀ ¼½ ?>Ê Á  ¾‹¹»¹"½=½¼ ½ ¼ º ¾‹¹"¹»º8º8¼ ºÄ … ÃÆÅuÇ=È¥É

Here Ë p  p Í̬ÎM!Ï defined similarly. Ø (FIM).

]ZÐ ÑÒ Ôo ÓMoÕ rUÖ p  t r and Ë  , Ë  and Ë  are p t t p t t × p] is Ï called the Fisher information matrix

IV. L OCALIZATION E RRORS WITH R ANGE M EASUREMENTS In this section we analyze the target localization when range measurements are used in the estimation and the error is proportional to the range. We first derive the CRLB for a fixed setup and then use this in deriving the a priori CRLB. We will compare the error bounds that we obtain with least square and weighted least square estimators. Recall from Section III-B that the density of $  is

_    n    U     where

— OÙ b ۗ Ú ÞÜ ÝC—ZÝ bb ß ]>à Üâá>—Má bß ] ] c ] ã ] ˆ ` X P¥z 

z äå{ Î }›vmæ   !   S vå  !   S ×

Since the measurement at each node is independent of those at the other nodes, we have

_    ^=^=^   n     ^=^^         T a a T T a kbÔê — Ù b — Ú Ü Ý:—ZÝ bOß ]>à Ü á>—Má bFß ] ] ^ c ]ã b]  ‡ ˆ ` T±ç T P ‰ Ôè z  "X é  a In the following, we will denote the above conditional density _ by From this we obtain the terms of the FIM, which we denote by ëË , for a given network to be _ R S q ò Z ó  ô  î ëË p  p ì!íÌ ’ Ö p  txïñð  S õ ö T  c v ` { S@ð :  !  F S  (6)  qè { S I  S }„v÷I  S _ a ëË p  t  Ë t  p ì!íÌ î ’ Ö p  txïñð SRòq óZô õ ö T  c v ` { S@ :  !  ð  = ð  !   ^ (7)  { S I S }„v÷I  S qè ^=^^ $ a œ . Ë t  t is identical to Ë p  p with   Here $ ™ øš $  T and  replaced aby   and  respectively. Recall that I8  æ   !  7 S vå  !  F S ^ We are now ready to perform the a priori analysis where we know only the distribution of the node locations rather than the actual locations. For this analysis, the observation space will include the location of the sensors in addition to the range measurements. Thus the observation space for the a priori ™ ú ü ™ where ü ™ ·š   C ^=^=^  T  T œ . analysis is ù    $û a a denote as Ë will The terms in the a priori FIM which we shall be

Ë p t 





_  $ ™  ü ™ n     = S q ò ä ÿ  î þ Ë t  p ý!íÌ ’  ’ Ö p  t ï˜ð   õ n _ ü     ð ð ™ : S Ô ò  ÿ   !íÌ þ ’ Ö p  txïñð   õ ð ð S:òÔÿ _  $ ™ n ü ™      !íÌ þ ’ Ö p  t ï Ì î ’ Ö þ ’ ï ð   õ±õ üË  v Ë$  ð ð p t p t

þ rr

]UÐ 

Here we have used the fact that !íÌ î ’  þ ’ Ö p  t ÎLÎLÏ p oÔÓMo t ’ ×R× Ï term is independent of the measurements. Note that the Ï first üË  will be the same for localization using either range p t measurements or using range-difference measurements. Ë p  p and Ë t  t can be similarly split into sum of two terms one of which is independent of the measurements. A. A Priori Error Bounds for C ‚ AM Using the joint density of ü definition of Ë Õ  Õ , we get

Ë ü p t  Ë ü p p 

ü™

α = 0.1 −3 K = 10 λ = 16 −4

10

−5

10

CRLB

given in Eqn. 4 and the

Ë ü t p  f Ë ü t  t  ` wN P

Ë$ p  t can be written as: _  $ ™ n ü™ : S Ô ò  ÿ  þ î þ Ë$ p  t ·!íÌ ’ Ö p  t ï Ì ’ Ö ’ ï ð   ð ð Observe that the inner expectation is the ‘  

−3

10

−6

10

−7

10

(8) −8

10

    

3

6

10

õ±õ

(9)

  ’ term of the

Fisher information matrix of a known network as given in Eqn. 6. Making a transformation from the Cartesian to polar coordinates I" Eqn. 9 becomes the

A

Fig. 2. The CRLB as a function of the number sensors used in localization for  . We use !"# and $ % .



the field. Hence the bound obtained is not a function of the target location. The bound obtained above is a lower bound for the localization error even when the set of sensors whose measurements are used are not the closest. To see this consider localization ^=^=^  . By a similar using the measurements from sensors   T analysis to the above, the terms of thea a priori Fisher information matrix can be shown to be Ë p  t  Ë t  p and

` =ó    qÿä   öT Ë$ p  t ìÌ þ ’ Ö p  t  ï  c v { { S SO }„ v   S õ

è a Similar expressions can be written for Ë$ p  p , Ë$ t  t and Ë$ t  p .   location of the Given the target coordinates   , the     . Define  ™ â isœ completely described by E G sensor   I

™   ^^=^   œ . Since   ^^=^   are inde0 &,) 9.-Q0 ¸ +"&,/ B 9  š š  ^=^=^   T and » ¹ 8 º ¼ º " ¹  ½ ¼ ½ B T T  Å Å'&(*)+ a uniformly a a pendent of the  , and are distributed i.i.d. random &,/ 

variables, taking expectations with respect to

Ë$ p  t  Ë $ t  p  Ë$ p  p  Ë$ t  t  

f Ì ’



gives

ö T  ` P T¥ c v ` { S   S õ

`  è ï { S O}„v a

Ë t p  f Ë t  t  ` wN P v ` NQPv  ` P T¥` c {

0132

-



?54 076

¸

+98:<;

^^=^

(10)

The joint density for ™ is obtained from Eqn. 3. Using Eqns. 8 and 10 the a priori terms of the Fisher information are

Ë p t  Ë p p 

30

No. of nodes (n)

ö T  ` P TR c v ` { S8 Ì ’  ï ` { S O}„v   Sõ

è ` öa T v  S { c  Ì ï  Sõ (11) O  „ } v  S è a Since  ü Ë p  t and  Ë üt   p are f and Ë p  p  Ë t  t , the a priori  p and t will be  Ý a  Ý . The joint density of ™ is independent of { , and hence the a priori CRLB is related ] to { only through s  ] . For small values of { , the a priori S  CRLB can be seena to be approximately proportional to { S . 



We have not been able to obtain a closed form expression for Ë p  p calculated above but we have numerical computations to support this. Note that since the target is present in a Poisson field, it is estimated with the same accuracy at all points in

9B À

The expectations are maximum when     S   T are the smallest possible. Therefore the a priori a CRLB obtained for the C ‚ AM will be a lower bound for the error when any set of sensors are used to localize  ü For the special case when }  f , the can be seen to be f . This is easy to see. The information provided by the sensor is inversely proportional to its distance from the target. The density function of the location of the first sensor though does not go to zero as the sensor gets closer to the target. Note that the Fisher information is the expected value of this information and hence becomes = . sensor is at least > If we make the assumption that  the >closest away from the target i.e, ë   v ¬ and ¬ has a distribution as in Eqn. 2. In this case, it can be > shown that the CRLB will be identical to Eqn. 11 with }  . The expectation in Eqn. 11 is numerically evaluated as the average over c f Y@? Poisson deployments of the network and is then used to obtain the CRLB. Figures 3 and 2 summarize the results obtained. Fig. 2 is a log-log plot of the a priori CRLB as a function of . The plot is a straight line with slope ! f ^ c indicating only a marginal improvement in the CRLB as the number of sensors used is increased. This behavior can be explained as follows. Since the error is proportional to the range, the Fisher information provided by the EOG sensor

−3

10

Parameter (α) K = 10−3 n=6

¹
¹\S 8º ¼ ½ Å ¹
0.00625 0.025 0.1

−4

10

Å

−5

10

CRLB (x)

Å

/



MZbc

6 +"8 a 6

6 +98 a 6

À ¾

À ¾

8 a 6 9 + 8 a

8 a 6 " + 8 a

À

ÀñÀ

Ê

−6

10

The a priori CRLB will then be

d 8 Ufe¿º

−7

10

Å

−8

10

0

1

10

2

10

10

Node density (λ)

Fig. 3. The CRLB as a function of B % .

$

for fixed



. We use

!A#

É

A

B9



MZbgc



6 +98 a 6

À ¾

8 a 6 +"8 a

À

É

?

(12) and

decreases as increases. Thus for large , the increase in the Fisher information on using an additional sensor is small. Fig. 3 is a log-log plot of the a priori CRLB as a function of N . The plot is a straight line with slope approximately equal to ! CS . From the two plots, it can be inferred that a denser deployment will provide a better improvement in the estimation than an increase in .

We now make the following observations. H For the case when h   jk i jÆj c , using a Taylor series expansion for òÔóMôL c v9h we can write the a priori CRLB as

In this model, the measurements from all sensors within a u from distance of the target are used to localize the target.  from Let ‚ be the number of sensors within a distance of the target. Following the same procedure as above we can write the terms of the a priori Fisher information matrix to be



¶LK #B MON Q0 P 0 S R 5TU R 5WVYX156Z979 K K ¹ 8º ¼ ½ Å ¾ G ¼ HJI 8º ¼ ½  4 D F 4 E E ¶ ¶ K B MZN Q0 P 70 V¬979 K B 1 MON 6 0QP 0,U R X,Vx5U1 Àñ56ZÀ 979 K K K K Å ¾ 4 D 4 G E ¼ HE I º8¼ ½ 1 6 + 1 6 K B MZN Q0 P  0 S R K X@K U R [5 Vx571 576Z979 + 1 6 À˜À 0 V 9 ¹ 8 º ¼ ½ ¹ \  U 8 º ¼ ½ ¹ < S 8 º ¼ ½ Å 4D + + expressed. The probability Ë p  p : Ë t  p and Ë t  t can be similarly _ mass function (pmf) of ‚ _ ,  ‚ n , and the conditional density ü ü   ˜ ‚ , are obtained from of ™ given  ‚  ,  ™ Eqns. 1 and 5 respectively. Since both are independent of the ü target location     , the Ë ‚ ŸqŸ and Ë UŸÔ=Ÿâ terms are zero. Arguing as in the derivation ü for C‚ AM and using Eqn. 6 and the conditional density of ™ given  ‚    x from Eqn. 5 we

ü



ü



p 



t

{ S` c ! h voö n  !Æc  h  YRa hS ï Y¥a 

NQPñ c v { S c vmh õ qè { S` { S }` S a  l  QN Pñ c v { S [ h S QN Pñ c v { S S i H For the case when jkq prp c , we can write the a priori l

CRLB as

B. A Priori Error Bounds for FRAM

get

d 8 Ufe¿½ B ¸0 / (*) +9&,/

Å



ü

ü t   { S` Y¥a l q ò Z ó ô NQPñ c v { S ï ï } õ ! c õ Eqn 12 does not hold for }  f . For this, as in 

p 

Remark 2: the case of C‚ AM, the CRLB can be shown to be zero. C. Error Comparisons

We compare the performance of the least square estimator (LSE) and the weighted least square estimator (WLSE) with the a priori lower bounds obtained above via simulations. These estimators are as below.

y

Here v are

{ {



 þJs*t x w Jþ s*t





  v L Qu fyu RY a u zy v Qu YRa u RY a u Y¥a

is the measurement error covariance matrix and u ú ` and ú c matrices defined as follows. > c  `   !   > `  `   a !   > c    S a !  S !   S v  S !   S v  S  a a a

,

The simulation experiments are performed as follows. For C ‚ AM we deploy sensors randomly around a target located at the origin with the locations drawn from the joint density of Eqn. 4. For FRAM we deploy the sensors in a circle of

0

−2

10

10

For λ = 16 α = 0.1 K = 10−3

λ= 16 α = 0.1 −3 K = 10

CRLB LS WLS

LSE WLSE CRLB

−1

10

−3

10 −2

error

Error

10

−3

10

−4

10

−4

10

−5

10

−5

3

5

10

15

20

25

30

10

0.7

0.8

0.9

1 Radius of activation (R0)

No. of nodes (n)

Fig. 4. Performance of LSE and WLSE against the a priori CRLB for C | AM as a function of B .

1.1

1.2

1.3

Fig. 6. Performance of LSE and WLSE against the a priori CRLB for FRAM as a function on ‚ a .

−2

10

For n=6 α = .1 K = 10−3

CRB LS WLS

of magnitude lower than the WLSE. Fig. 6 is a log-linear plot of the error in the estimate as   . It is seen that as   increases the the a function of LSE performance improves only marginally while the WLSE continues to improve. This is similar to the behavior exhibited by the WLSE and LSE for the C ‚ AM as is increased but the error magnitudes are different.

−3

Error

10

V. L OCALIZATION WITH R ANGE -D IFFERENCE M EASUREMENTS

−4

10

−5

10

2

10

64

Node density (λ)

Fig. 5. Performance of LSE and WLSE against the a priori CRLB for C | AM as a function on $ .

u

radius around the target located at the origin according to a Poisson process. For each deployment we obtain the measurements from the sensors as Gaussian random variables with mean equal to the distance from the origin and variance C proportional to the distance with }  c f Y . The value of { ^ is taken as f c as ranging errors using RSS is around c f*} ( [7]). The measurements are then used in the estimators above. The sample average of the mean square estimation error from c f ? samples are obtained for each parameter combination. As expected the WLSE performs better than the LSE. Fig. 4 is a log-linear plot of the error in the estimate of the  coordinate as a function of . Beyond ~ , the improvement in the performance of the LSE is marginal while the WLSE continues to shows some improvement till € f . Fig. 5 is a log-log plot of error in the estimators as a function of N . The plots for the error in LSE, error in WLSE and the CRLB are all straight lines with slope SC but the CRLB is almost a order

We now obtain the a priori CRLB for localization with range-differences rather than the range measurements. The analysis is similar to that in Sec. IV for range based localization and only the results are summarized here. Recall that the range-difference measurements are modeled as jointly Gaussian with covariance matrix † . Define JL  as the >#MEGZ element of † YRa . The following may be obtained as before. H The terms of the Fisher information matrix for a given network are

¹ ƒ º8¼ ½ Å

¹ ƒ ½¼ º -2 ? -2 ¼ † Ņ„: 0 É ? †0 É : - ? ¹ ƒ º8¼ º Ņ„ : ¼ † 0 2 É 0 2 É : ? † -2 ? -2 »¹ ƒ ½=¼ ½ Ņ„: ¼ † 0 É 0 É : ? †

? ? ?

? ?

?

0ˆ‡ b‰*Š :\‹ ? ¾

‡ b‰*Š

? 90 ‰[ŒONŠ :\‹ ? ¾

‰[ŒONŠ

? 9 5

0ˆ‡ b‰*Š :\‹ ? ¾

‡ b‰*Š

? 90ˆ‡ b‰*Š :\‹ ? ¾

‡ bg‰Š

? 9 5

0 ‰[ŒONŠ :\‹ ? ¾

‰[ŒONŠ

? 90 ‰[ŒZN`Š :<‹ ? ¾

‰[ŒZNŠ

?9 (13)

Remark 3: From the above expression, we can see that the a priori CRLB will not be affected by the closeness of the sensors to the target. Hence for the error model considered, the lower bound will be the same for any set of sensors that are used to localize the target.

H

First consider C‚ AM. The terms of the a priori Fisher information matrix for C ‚ AM are

Ë p t  Ë t p  Ë p p  Ë t t 

fw ` NQPv T ö ¥Y a J è a



_

ü 

H

ü 

p 

t 

` NQPv T ö ¥Y a J è a

_

{

Y¥a ^ 

We consider two special cases. If the measurement errors are independent of each other, the covariance matrix † is of the form zRS where z is the variance in the measurements and  is the identity matrix. In this case, the a priori CRLB reduces to



ü

ü 

p 

` t  ï NwP v

! c Y¥a zS õ

[25] describes the following covariance matrix structure for the case of dependent measures.

`

S Ž †  zR`o  

c c ^^ ^=^=^^ ` c c c c ` ^^=^ .. .

.. .

.. .

..

‘,’ ’ ’

(14)

“

o T YRa r ” o T RY a r

.

For this case the CRLB is

ü 

H

p 

` c U S YRa t ï ` NQPv  !d zS õ

ü 

For FRAM, the terms of the a priori Fisher information matrix are

Ë p t  Ë t p  Ë p p  Ë t t 

fw Ì•



and the a priori CRLB is

H



ü

p 

ü 

t 



Ì

T ö RY a J è a

•



_

T ö RY a J è a



_

^ Y¥a

r

If the measurement errors are independent of each other, the a priori CRLB is



ü



p 

ü

Rz S t  ONQP  S !dc

and if they are dependent with a covariance matrix as above, the CRLB is given by



ü

p 



ü t

l

We compare the performance of the plane intersection (PI) method discussed in [26] with the a priori calculated CRLB. PI transforms the location estimation problem with range-difference measurements into a least square estimation {  problem ofœ2 the form of o   ” r ” – r o T Y¥a S S a T Y¥a ” a where  ·š   and

{

and the a priori CRLB will be



A. Error Comparisons

` ONQP  Rz SS ! c

 c   `   c 

  v  a S v c` a ‡  S  aS v  

From this the estimator is

  {

    S  a S a  S a  S v a S  {

YRa

v   S v    aS a v  S S v  S v  a   S aS {  

S a S v  S ‰

We consider c f ? deployments of the network. For each deployment we obtain the range-difference measurements from the sensors as a jointly Gaussian random variables with means equal to the range-difference and covariance matrix of ^ f `˜— . The sample the form defined in Eqn. 14 with z  average of the mean square estimation error is obtained for each parameter combination. Fig. 7 shows the semi log plot comparing CRLB with PI estimator for Joint Gaussian error model. Since the structure of the PI is also a least square estimator, it exhibits a behavior similar to the LSE considered for range measurements. In the region of interest i.e., at ™~ the CRLB is about c f times lower than the error in the PI. Also, it is seen that CRLB, unlike in the case of localization using range, shows a improvement as the number of sensors used increases. This behavior can be explained as follows. Since the error in the range-difference measurement have a fixed variance, with each extra sensor the Fisher information increases by a fixed amount. Fig. 8is semi log plot of the  isa varied. performance of the PI estimator as From the two plots, we see that if the in C‚ AM is taken as the average  from the target, the number of sensors within a distance of CRLB for the two model are approximately equal. However this not true for the PI estimation error. Remark 4: For j c f , the bound for localization using range measurements is smaller while for p c f the bound is smaller when range-difference measurements are used. This is due to the difference in the nature of the error models assumed for the two cases. VI. D ISCUSSION

AND

C ONCLUSION

In this paper, the a priori CRLB for different localization setups as a function of network parameters was calculated. From simulations and analytical analysis we could show that H For the case of localization using range measurements, the decrease in the CRLB as more sensors are used  ) will be marginal. The (directly or by increasing dependency is stronger on N . H The bound will be the smallest when sensors closer to the target are used.

0

10

Plane Intersection CRB

λ = 16 σ =0.025 −1

10

ITs

−2

var(x)

10

IT

−3

10



S

−4

10

 s T

−5

10



5

10

15 20 Nearest Nodes (N)

25

30

T



Fig. 7. C | AM : Performance of the PI estimator against the a priori CRLB for CNAM as a function of B . −1

10

PI CRLB λ = 16 σ = 0.025 −2

10

−3

Fig. 9. Illustrating the notation used in the derivation of the probability that B closest nodes are not those with the B least measurements.

−4

A PPENDIX

var(x)

10

10

A. Identification of the −5

10

−6

10

0.35

0.4

0.45

0.5

0.55 0.6 0.65 0.7 Radius of Activation (R)

0.75

0.8

0.85

Fig. 8. C | AM : Performance of the PI estimator against the a priori CRLB for FRAM as a function of ‚ a .

H

The probability that the closest sensors will not have the smallest reading is significant and so noisy measurements cannot be used to identify the closest sensors. H In the case of range-difference measurements, the CRLB will decrease when both the density of deployment and the number of sensors used are increased. H For localization using range-difference, the lower bound, under the error model assumed, will not depend on the closeness of the sensors to the target. H The a priori error in LS, WLS and PI estimators in the region of interest are almost a order of magnitude higher than the CRLB. H The improvement in performance as more sensors are used is greater for WLSE than LSE. The error in range measurements can also be modeled as a Gamma distribution function. Though this model has the advantage of being one sided, it cannot be used to account for errors in the knowledge of node locations i.e., } .

Sensors Closest to the Target

We show that the probability that -closest will not produce the lowest range measurements is significant by calculating the probability that the  v WšQ› sensor has a reading that is smaller than the šQ› sensor. % Let   be the event that $ s  is smaller than $  and T % let œ4  denote the probability of   . œ4  can be written as

œ4

 



zž Ÿ¢¡ zž Ÿ¢¡

F$ T s  j $ T L   ‹ j { t  v



v

{ t

  ^

Here   is the area of the annulus between the concentric {   circles with center as target and radii  and o Ôs  r . t U is the area of the annulus the circles with center at   andbetween $ the target and radii (see Fig. 9).  { `  % as the The area t  can be approximated by P % T T error is generally small compared { t to the range T . Using T  v we can write œ4  a similar approximation for as

 £fž<ŸJ¡ d ‹j ` P  T % T v ` P  T s  % T s  (15)  j  s  and ¤¦¥ ž  % j Since the random variable % T T T ¤§¥ ž  T s  we can write Eqn. 15 as: J   ‹¨ h œ4  ‹h Jä v  j ©ZP  T % T . We first evaluate where J   is zž Ÿ¢¡    ‹ª Jä%   conditioned on which we shall denote by J    . T  T given T is a zero mean Gaussian with variance { }÷vI T œ4

0.45

0.45

k=1 k=2 k=3 k=4 k=5

0.4

RSS TOF 0.4

0.35

0.3

0.3

0.25

0.25

p(n,1)

0.35

p(n,k) 0.2

0.2

0.15

0.15

0.1

0

0.1

K = 10e −3 α = .05 λ=8

0.05

0

2

4

6

8

10

12

14

16

λ=8 α = .5 RSS α = .0625 TOF K = 10e −3

0.05

18

0

20

0

5

10

15

20

Fig. 10.



while d



J  

Plot of

«¬ B®­°¯²±

vs

B

for different values of

¯

Fig. 11.





is a šQ› order Erlang (inter-arrival area). Therefore becomes

‘ ´ °] µ ]–¶FkT] · á ] ] ÜO¸ à˜¶ k ß ] Y “ J     ³ nè  ³ èn  ˆ X ` C p t p Ž © P Ž S I T { } v÷I T §¹    p  ú ï N  Y¥!da Xc YLV [ õ ¹   c»º ` !m³ nè  gŒ ¼#½ ï ©MP { I O}„v…I õ p  pT  T N X ¥ Y a L Y [ (16) ú ï  !dc V õ ¹ A common approximation for gŒ ¼#½   which is correct upto

two decimal places is:

¾



Á lÀ¿

^  ^  ^ ©c à 7© © ! ^— f

for for for

 ` ^` fx ` ^ `h h  h h ` ^ ~ ` ^~ h  ™ h =

25

30

35

40

45

50

Sensor Index n

Sensor Index n

(17)

Using the above approximation and the density function for  given in Eqn. 2 we can uncondition J    to obtain Jä T  . A closed form does not exist for Jä  and we evaluate this numerically for suitable values of N ,r{ Ä`Ä and K. Typically, the range measurement obtained from has a error of — about } while the range measurement obtained from ÅuùFÆ is more accurate and has an error a order of magnitude lesser ^— [7]. Therefore we take { as for range measurements derived rÄ`Ä  — ` ^ from and f ~ for C range measurements derived from ÅuùFÆ . We fix } at c f Y . Fig. 10 gives the plot of J   rvs Ä`Ä for different values of when the ranging is done using 11 plots the value rÄ`Ä and Åu. ùFFig. of Jä for ranging using on the same plot. Æ As expected, Jä  is higher when ranging is done using FÄÄ . As seen from the above plots, Jä  and hence œ4  is significant for all . From this we may conclude that proba-

Plot of

«5¬ BJ­°¯»±

vs

B

for ranging using RSS and TOF

bility that the -closest sensor get the is significant.

smallest measurement

R EFERENCES [1] I. F. Akyildiz et al, “Wireless sensor networks: A survey,” Computer Networks: The International Journal of Computer and Telecommunications Networking, vol. 38, pp. 393–422, 2002. [2] F. Zhao and L. Guibas, Wireless sensor networks: An Information Processing Approach, Elsevier, 2005. [3] J. O. Smith and J. S. Abel, “Closed-form least-squares source location estimation from range-difference measurements,” IEEE transactions on Acoustics, Speech and Signal Processing, vol. 35, no. 12, pp. 1661–1670, December 1987. [4] Y. Huang and et al, “Real-time passive source localization:A practical linear-correction least-squares approach,” IEEE transactions on Speech and Signal Processing, vol. 9, no. 8, pp. 943–957, November 2001. [5] P. A. P. Moran, “Estimating structural and functional relationships,” Journal of Multivariate Analysis, 1970. [6] A. Karnik and A. Kumar, “Iterative localisation in wireless ad hoc sensor networks: One-dimensional case,” in Proceedings of IEEE-INFOCOM, 2005. [7] A .Savvides, W. L. Garber, R. L. Moses, and M. B. Srivastava, “An analysis of error inducing parameters in multihop sensor node localization,” IEEE transaction on Mobile Computing, 2004. [8] N. Patwari, R. .J. O’Dea, and Y.Wand, “Relative location in wireless networks,” in Proceedings of IEEE - Vehicular Technology Conference, 2001. [9] N. Bulusu, J. Heidemann, and D. Estrin, “GPS-less low cost outdoor localization for very small devices,” IEEE Personal Communication Magazine, 2000. [10] L. Doherty, K. S. J Pister, and L. .E. Ghaoui, “Convex position estimation in wireless sensor network,” in Proceedings of IEEEINFOCOM, 2001. [11] N. Sundaram and P. Ramanathan, “Connection based location estimation scheme for wireless adhoc networks,” in IEEE Global Communication Conference, 2000. [12] D. Niceluscu and B.Nath, “Ad-hoc positioning system,” in IEEE Global Telecommunication Conference, 2001. [13] R. L. Moses, D. Krishnamurthy, and R. M. Patterson, “A self localization method for wireless sensor networks,” EURASIP Journal on Applied Signal Processing, pp. 348–358, 2003. [14] A. Savvides, S. Han, and M. .B. Srivastava, “Dynamic fine grained localization in ad hoc networks,” in 7th ACM Annual International Conference on Mobile Computing and Networking, 2002. [15] J. Albowicz, A. Chen, and L. Zhang, “Recursive position estimation in sensor networks,” in IEEE -International Conference on Network Protocols, 2001.

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On the A Priori Localization Errors in Sensor Networks

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