IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 8 ...ece.utah.edu/~npatwari/pubs/patwari03-print.pdf · 2138 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 8, AUGUST 2003

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 8, AUGUST 2003 2137

Relative Location Estimation inWireless Sensor Networks

Neal Patwari, Member, IEEE, Alfred O. Hero, III, Fellow, IEEE, Matt Perkins, Member, IEEE,Neiyer S. Correal, Member, IEEE, and Robert J. O’Dea, Member, IEEE

Abstract—Self-configuration in wireless sensor networks isa general class of estimation problems that we study via theCramér–Rao bound (CRB). Specifically, we consider sensorlocation estimation when sensors measure received signal strength(RSS) or time-of-arrival (TOA) between themselves and neigh-boring sensors. A small fraction of sensors in the network havea known location, whereas the remaining locations must beestimated. We derive CRBs and maximum-likelihood estimators(MLEs) under Gaussian and log-normal models for the TOA andRSS measurements, respectively. An extensive TOA and RSSmeasurement campaign in an indoor office area illustrates MLEperformance. Finally, relative location estimation algorithms areimplemented in a wireless sensor network testbed and deployed inindoor and outdoor environments. The measurements and testbedexperiments demonstrate 1-m RMS location errors using TOA,and 1- to 2-m RMS location errors using RSS.

Index Terms—Cramér–Rao bound, localization, radio channelmeasurement, self-configuration, sensor position location estima-tion, signal strength, time-of-arrival.

I. INTRODUCTION

WE CONSIDER location estimation in networks in whicha small proportion of devices, called reference devices,

havea priori information about their coordinates. All devices,regardless of their absolute coordinate knowledge, estimate therange between themselves and their neighboring devices. Suchlocation estimation is called “relative location” because therange estimates collected are predominantly between pairs ofdevices of which neither has absolute coordinate knowledge.These devices withouta priori information we call blindfoldeddevices. In cellular location estimation [1]–[3] and localpositioning systems (LPS) [4], [5], location estimates are madeusing only ranges between a blindfolded device and referencedevices. Relative location estimation requires simultaneousestimation of multiple device coordinates. Greater locationestimation accuracy can be achieved as devices are addedinto the network, even when new devices have noa prioricoordinate information and range to just a few neighbors.

Manuscript received October 3, 2002; revised April 7, 2003. This work wassupported in part by a National Science Foundation Graduate Research Fellow-ship for N. Patwari and by ARO-DARPA MURI under Grant DAAD19-02-1-0262. The associate editor coordinating the review of this paper and approvingit for publication was Dr. Athina Petopulu.

N. Patwari and A. O. Hero, III are with the Department of Electrical Engi-neering and Computer Science, University of Michigan, Ann Arbor, MI 48104USA (e-mail: [email protected]; [email protected]).

M. Perkins, N. S. Correal, and R. J. O’Dea are with Motorola Labs, Planta-tion, FL, USA (e-mail: [email protected]; [email protected];Bob.O’[email protected]).

Digital Object Identifier 10.1109/TSP.2003.814469

Emerging applications for wireless sensor networks willdepend on automatic and accurate location of thousands ofsensors. In environmental sensing applications such as waterquality monitoring, precision agriculture, and indoor air qualitymonitoring, “sensing data without knowing the sensor locationis meaningless”[6]. In addition, by helping reduce configurationrequirements and device cost, relative location estimation mayenable applications such as inventory management [7], intru-sion detection [8], traffic monitoring, and locating emergencyworkers in buildings.

To design a relative location system that meets the needsof these applications, several capabilities are necessary. Thesystem requires a network of devices capable of peer-to-peerrange measurement, anad-hoc networking protocol, and adistributed or centralized location estimation algorithm. Forrange measurement, using received signal strength (RSS) isattractive from the point of view of device complexity and costbut is traditionally seen as a coarse measure of range. Time-of-arrival (TOA) range measurement can be implemented usinginquiry-response protocols [7], [9]. In this paper, we will showthat both RSS and TOA measurements can lead to accuratelocation estimates in dense sensor networks.

The recent literature has reflected interest in location esti-mation algorithms for wireless sensor networks [8], [10]–[16].Distributed location algorithms offer the promise of solvingmultiparameter optimization problems even with constrainedresources at each sensor [10]. Devices can begin with localcoordinate systems [11] and then successively refine theirlocation estimates [12], [13]. Based on the shortest path froma device to distant reference devices, ranges can be estimatedand then used to triangulate [14]. Distributed algorithms mustbe carefully implemented to ensure convergence and to avoid“error accumulation,” in which errors propagate serially in thenetwork. Centralized algorithms can be implemented when theapplication permits deployment of a central processor to per-form location estimation. In [15], device locations are resolvedby convex optimization. Both [8] and [16] provide maximumlikelihood estimators (MLEs) for sensor location estimationwhen observations are angle-of-arrival and TOA [8] and whenobservations are RSS [16].

In this paper, we mention only briefly particular location es-timation algorithms. Instead, we focus on the accuracy pos-sible usingany unbiased relative location estimator. The radiochannel is notorious for its impairments [17], [18], and thus,sensor location accuracy is limited. The Cramér–Rao bounds(CRBs) presented in this paper quantify these limits and allow

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2138 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 8, AUGUST 2003

determination if the accuracies necessary for particular applica-tions are possible.

We begin in Section II by considering CRBs for networkself-calibration estimators. Next, we state the relative locationestimation problem and derive CRBs and MLEs in Section III.In Section IV, measurements of TOA and RSS are used to il-lustrate estimator performance. Finally, real-time operation ofrelative location is demonstrated in Section V. Photos of the ex-periments are included in an extended electronic version of thispaper in [19].

II. NETWORK ESTIMATION BOUNDS

In network self-calibration problems, parameters of all de-vices in a network must be determined. Information comes bothfrom measurements made between pairs of devices and a subsetof devices that knowa priori their parameters. A network self-calibration estimator estimates the unknown device parameters.For example, distributed clock synchronization in a networkcould be achieved by devices observing pair-wise timing off-sets when just a small number of devices are synchronous.

Specifically, consider a vector of device parameters. Each device has one parameter. Devices

are blindfolded devices, and devicesare reference devices. The unknown parameter vector is

, where for . Note thatare known. Devices and make

pair-wise observations with density .We allow for the case when devices make incomplete ob-servations since two devices may be out of range or havelimited link capacity. Let device makes pair-wiseobservations with device . By convention, a device cannotmake a pair-wise observation with itself, so that . Bysymmetry, if , then .

We assume by reciprocity that ; thus, it is suf-ficient to consider only the lower triangle of the observationmatrix when formulating the joint likelihoodfunction. In practice, if it is possible to make independent ob-servations on the links fromto and from to , then we as-sume that a scalar sufficient statistic can be found. Finally, weassume that are statistically independent for . Thisassumption can be somewhat oversimplified (see [20] for theRSS case) but necessary for analysis. Using measurements likethose in Sections IV and V remains important to verify true per-formance. The log of the joint conditional pdf is

(1)

The CRB on the covariance matrix of any unbiased estimatoris cov , where the Fisher information matrix (FIM)

is defined as

......

... (2)

As derived in Appendix A, the diagonal elements ofreduce to a single sum over since there are

terms in (1) that depend on . The off-diagonal elementscan be further reduced: When , there is at most one sum-mand in (1) that is a function of bothand ; thus

(3)

where is an indicator function: 1 if or 0otherwise.

A. Conditions for a Decreasing CRB

Intuitively, as more devices are used for location estimation,the accuracy increases for all of the devices in the network. Foran -device network, there are parameters but vari-ables used for their estimation. The analysis of this sec-tion gives sufficient conditions to ensure the CRB decreases asdevices are added to the network. Consider a network ofblind-folded devices and reference devices. Now, consider addingone additional blindfolded device. For theand blind-folded device cases, let and be the FIMs defined in (2),respectively.

Theorem 1: Let be the upper left blockof . If for the blindfolded device case 1)

, and 2)device makes pair-wise observations between itself andat least one blindfolded device and at least two devices, in total;then two properties hold: 1) in the positivesemi-definite sense and 2) tr tr .

Theorem 1 is proven in Appendix B. The Gaussian and log-normal distributions in Section III meet condition 1). Property1) implies that the additional unknown parameter introduced bythe st blindfolded device does not impair the estimationof the original unknown parameters. Furthermore, property 2)implies that the sum of the CRB variance bounds for theun-known parameters strictly decreases. Thus, when a blindfoldeddevice enters a network and makes pair-wise observations withat least one blindfolded device and at least two devices in total,the bound on the average variance of the originalcoordinateestimates is reduced. Note that properties 1) and 2) of Theorem1 would be trivially satisfied by the data processing theorem ifadding a device into the network did not increase the number ofparameters.

III. RELATIVE LOCATION ESTIMATION

In this section, we specialize for device location estimationusing pair-wise RSS or TOA measurements in a wirelessnetwork. Specifically, consider a network of referenceand blindfolded devices. The device parameters

, where, for a two-dimensional (2-D) system,(although extension of these results to 3-D is

also possible). The relative location problem corresponds to theestimation of blindfolded device coordinates

(4)

given the known reference coordinates. In the TOA case, is the mea-

sured TOA between devicesand in (seconds), and in the

PATWARI et al.: RELATIVE LOCATION ESTIMATION IN WIRELESS SENSOR NETWORKS 2139

RSS case, is the measured received power atdevice transmitted by device (in milliwatts). As discussedin Section II, only a subset of devices make pair-wisemeasurements with device, and aretaken to be upper triangular matrices, and these measurementsare assumed statistically independent.

In addition, assume that is Gaussian distributed withmean and variance , which is denoted

(5)

where is the speed of propagation, and is not a functionof . We assume that is log-normal; thus, the randomvariable dBm is Gaussian

dBm dBm

dBm dBm (6)

where (dBm) is the mean power in decibel milliwatts,is the variance of the shadowing, and (dBm) is the receivedpower in decibel milliwatts at a reference distance. Typically,

m, and is calculated from the free space path lossformula [21]. The path loss exponent is a function of the en-vironment. For particular environments, may be known fromprior measurements. Although we derive the CRB assumingis known, it could have been handled as an unknown “nuisance”parameter.

Given (6), the density of is

(7)

Here, is the MLE of range given received power .Neither nor are assumed to be ergodic random

variables; in fact, obstructions in the measured environment thatcause shadowing and TOA errors do not usually change overtime. The CRB gives a lower bound on the ensemble varianceover different random shadowing environments. If networkswith the same relative device coordinates are implemented inmany different areas, the variances of any unbiased coordinateestimator will be lower bounded by the CRB presented here.

The model assumptions made in this section will justifiedby experiment in Section IV. In the next sections, we use thesemodels to derive the CRB and MLE for both RSS and TOA mea-surements.

A. One-Dimensional TOA Example

Consider using TOA measurements to locate devices thatare limited to a one-dimensional (1-D) linear track. This could,for example, be applied to location estimation on an assemblyline. Consider blindfolded devices and reference deviceswith combined parameter vector . Theunknown coordinate vector is . Assume

all devices make pair-wise measurements with every otherdevice, i.e., . Thedistribution of the observations is given by (5) with

. The second partials of are, , which are constant

with respect to the random variables . Thus, the FIM, whichis calculated using (3), is ,where is the identity matrix, and is an by 1 vectorof ones. For , the matrix is invertible

The CRB on the variance of an unbiased estimator foris

(8)

Expression (8) implies that the variance is reduced morequickly by adding reference () than blindfolded ( ) devices.However, if is large, the difference between increasingand

is negligible.

B. Two-Dimensional Location Estimation

In the remainder of this article, we focus on 2-D location es-timation of (4). We denote by and the FIMs for the RSSand TOA measurements, respectively. Each device has two pa-rameters, and we can see that the FIM will have a similar formto (2) if partitioned into blocks

(9)

where and are given by (2) using only theparam-eter vector , and and are given by (2) usingonly . The off-diagonal blocks and are simi-larly derived. The elements of the submatrices of (9) are derivedin Appendix C. For the case of RSS measurements, the elementsare given by

(10)


For the case of TOA measurements, the elements are

(11)

Note that while . These SNRquantities directly affect the CRB. For TOA measurements, thedependence on the device coordinates is in unit-less distance ra-tios, indicating that the size of the system can be scaled withoutchanging the CRB as long as the geometry is kept the same.However, in the case of RSS measurements, the variance boundscales with the size of the system even if the geometry is kept thesame due to the terms in the denominator of each term of .These scaling characteristics indicate that TOA measurementswould be preferred for sparse networks, but for sufficiently highdensity, RSS can perform as well as TOA.

Let and be unbiased estimators of and . For thecase of TOA measurements, the trace of the covariance of theth location estimate satisfies

tr cov

(12)

For RSS measurements, replace in (12) with . For thecase of one blindfolded device, a simple expression can be de-rived for both RSS and TOA measurements.

C. Single Unknown Location Example

Consider the network having blindfolded device 1 and refer-ence devices . This example, with a single pair ofunknowns and , is equivalent to many existing locationsystems, and a bound for the variance of the location estimatorhas already been derived in for TOA measurements [2]. In thecase of RSS measurements

from which we obtain

where the distance is the shortest distance from the pointto the line segment connecting deviceand device .

For the case of TOA measurements, we obtain

(13)

The ratio has been called the geometricconditioning of device 1 w.r.t. referencesand [2].is the area of the parallelogram specified by the vectors fromdevice 1 to and from device 1 to, normalized by the lengthsof the two vectors. The geometric dilution of precision (GDOP),which is defined as , is

GDOP

which matches the result in [2]. The CRBs are shown in Fig. 1when there are four reference devices located in the corners ofa 1 by 1 m square. The minimum of Fig. 1(a) is 0.27. Since theCRB scales with size in the RSS case, the standard deviation ofunbiased location estimates in a traditional RSS system oper-ating in a channel with is limited to about 27%of the distance between reference devices. This performancehas prevented use of RSS in many existing location systemsand motivates having many blindfolded devices in the network.Note that in the TOA case, is proportional to , and thus,

was chosen in Fig. 1(b).

D. Maximum Likelihood Relative Location Estimation

For general and , we calculate the MLE of . In the caseof TOA measurements, the MLE is

(14)

where . The MLE for the RSS case is [16]

(15)

Unlike the MLE based on TOA measurements, the RSS MLE isreadily shown to be biased. Specifically, for a single referenceand single blindfolded device, the range estimate between thetwo devices is . Using (7), the mean of is given by

where

(16)


(a)

(b)

Fig. 1. � (in meters) for the example system versus the coordinates of thesingle blindfolded device for (a) RSS with� =n = 1:7 or (b) TOA withc� = 1 m.

For typical channels [21], , adding 20% bias to therange. Motivated by (16), a bias-reduced MLE can be definedas

(17)

However, there remains residual bias. Consider and. Place the reference devices at the corners of a 1 by 1 m

square and the blindfolded device within the square, the same asthe case plotted in Fig. 1. We calculate via simulation [22] thebias gradient norm of given by (17) and display it in Fig. 2.

The gradient of the bias can be used in the uniform CRB tocalculate the achievable variance of the biased estimator [22] ascompared with all other estimators with same or less bias gra-dient norm. Fig. 2 shows that the bias gradient is high (withnorm ) at the corners of the square. Expression (17) showsthat the MLE tries to force the ratio close to 1.If the blindfolded device is very close to one reference deviceand far away from the others, then measurements from the other

Fig. 2. Bias gradient norm of the RSS MLE ofx from (17) for the examplesystem of Section III-C.

three reference devices provide relatively little information re-garding the placement of the blindfolded device. In the limit asthe blindfolded device approaches a reference device, it can onlybe localized to a circle around that reference. Thus, no unbiasedestimator is possible. The MLE in (17) approaches a constant inthe limit, and thus, the bias gradient norm approaches 1.

IV. CHANNEL MEASUREMENTEXPERIMENT

In this section, we describe the measurement system and ex-periment and validate the channel model assumptions made atthe beginning of Section III. A set of multipoint-to-multipoint(M2M) wideband channel measurements were conducted at theMotorola facility in Plantation, FL. The measurement environ-ment is an office area partitioned by 1.8-m-high cubicle wallswith hard partitioned offices, external glass windows, and ce-ment walls on the outside of the area. There are also metaland concrete support beams within and outside of the area. Of-fices are occupied with desks, bookcases, metal and woodenfiling cabinets, computers, and equipment. Forty-four device lo-cations are identified within a 14 by 13 m area and marked withtape.

The measurement system uses a wideband direct-sequencespread-spectrum (DS-SS) transmitter (TX) and receiver (RX)(Sigtek model ST-515). The TX and RX are battery-poweredand are placed on carts. The TX outputs an unmodulatedpseudo-noise (PN) code signal with a 40-MHz chip rate andcode length 1024. The center frequency is 2443 MHz, and thetransmit power is 10 mW. Both TX and RX use 2.4-GHz sleevedipole antennas kept 1 m above the floor. The antennas have anomnidirectional pattern in the horizontal plane and a measuredantenna gain of 1.1 dBi. The RX records I and Q samples ata rate of 120 MHz, downconverts, and correlates them withthe known PN signal and outputs a power-delay profile (PDP).An example PDP is shown in Fig. 3. We ensure that noise andISM-band interference is not an issue by maintaining an SNR

dB throughout the campaign.For TOA, wireless sensors will likely make two-way

(round-trip) measurements [7], [9] due to the impracticalityof accurately synchronizing all devices in wireless sensor


Fig. 3. (a) Measured PDP with TX at 1 and RX at 24 and threshold (- - -)above which received power is integrated to calculate RSS. (b) Leading edge ofsame PDP showing LOS TOA= d =c (� - � - �) and estimated TOA (- - -).(c) Autocorrelation of PN signalR (�) used in template-matching [23].

networks. Two-way TOA measurements do not require syn-chronized devices since the round-trip delay can be measuredat a single device and then divided by two to estimate theone-way propagation time. However, for the purpose of thesemeasurements, two-way TOA measurements are not necessary.Instead, we carefully synchronize our TX and RX usingoff-the-shelf time-synchronization equipment. Errors due tomultipath are the predominant source of variance in TOAestimates: for the two-way TOA case this has been reported in[9], and for these measurements, this is demonstrated below.One-way synchronized and two-way TOA measurements areequally affected by the multipath channel; thus, this one-waymeasurement experiment closely approximates the TOAs thatwould be measured in a sensor network.

Both TX and RX are synchronized by 1 pulse per second(1PPS) signals from two Datum ExacTime GPS and rubidium-based oscillators. On each of the eight days of the campaign,a procedure is followed to ensure a stable time base. After aninitial GPS synch of the ExacTimes, GPS is disconnected, andthe rubidium oscillators provide stable 1PPS signals. The fre-quencies of the two rubidium oscillators are off very slightly;thus, the 1PPS signals drift linearly, on the order of nanosecondsper hour. By periodically measuring and recording the offset be-tween the two 1PPS signals using an oscilloscope, the effect ofthe linear drift can be cancelled. A time base with a standard de-viation of between 1-2 ns is achieved. The variance of the timebase ( ns ) is thus a small source of error in the measuredTOA variance (37 ns) reported in Section IV-A.

The M2M measurements are conducted by first placing theTX at location 1, while the RX is moved, and measurementsare made at locations 2 through 44. Then, the TX is placed atlocation 2, as the RX is moved to locations 1 and 3 through 44.At each combination of TX and RX locations, the RX recordsfive PDPs. Since we expect reciprocity, there are a total of 10measurements for each link. All devices are in range of all otherdevices. Over the course of the eight-day campaign, a total of

measurements are taken.

A. Estimating TOA and RSS

The wideband radio channel impulse response (CIR) is mod-eled as a sum of attenuated, phase-shifted, and time-delayedmultipath impulses [18], [21]. The PDP output of the Sigtekmeasurement system, due to its finite bandwidth, replaces eachimpulse of the CIR with the autocorrelation function of the PNsignal , which is shown in Fig. 3(c), an approximatelytriangular peak that is ns wide. In high SNR,low multipath cases, TOA estimates can be more accurate than

. However, a wider peak permits more multipath errorssince the line-of-sight (LOS) component, with TOA , canbe obscured by non-LOS multipath that arrive s afterthe LOS TOA. If the LOS component is attenuated, it can be dif-ficult to distinguish the LOS TOA. In Fig. 3(a), the PDP is seento contain several multipath within the first 200 ns. Inspectingthe PDP immediately after , as shown in Fig. 3(b), theLOS path at 42 ns is visible but attenuated compared with a latermultipath, which appears to arrive at 80 ns.

The template-matching method [23] provides a TOA estima-tion algorithm that is robust to such attenuated-LOS multipathchannels. In template-matching, samples of the leading edge ofthe PDP are compared with a normalized and oversampled tem-plate of , which is shown in Fig. 3(c). The TOA esti-mate is the delay that minimizes the squared-error betweenthe samples of the PDP and the template. In Fig. 3(b), the tem-plate-matching TOA estimate ns is in error by 9ns. If a local maximum was necessary to identify the LOS path,the error would have been much greater.

Since non-LOS multipath are delayed in time, usuallyhas a positive bias. We estimate the bias to be the average of

, , which in these measurements is 10.9 ns.In this paper, we assume this bias is known for environments ofinterest, however, and, similarly to , this bias could be esti-mated as a “nuisance” parameter. Subtracting out the bias fromour measurements, we get the unbiased TOA estimator. Fi-nally, the average of the 10 measurements for the link be-tween and is called . The measured standard deviation

is 6.1 ns.It has been shown that a wideband estimate of received power

can be obtained by summing the powers of the multipath inthe PDP [21]. To distinguish between noise and multipath, onlypower above a noise threshold is summed, as shown in Fig. 3(a).This wideband method reduces the frequency-selective fadingeffects. The geometric mean of the 10 measurements forthe link between and , which we call , reduces fading dueto motion of objects in the channel. Shadowing effects, whichare caused by permanent obstructions in the channel, remainpredominant in since the TX and RX are stationary. Shad-owing loss is often reported to be a log-normal random variable[18], [21], [24], which lead us to propose the log-normal shad-owing model in (6). As shown in Fig. 4, The measuredmatch the log-normal shadowing model in (6) withand dB, using m. The low variance may bedue to the wide bandwidth, averaging, and homogeneity of themeasured cubicle area.

We verify the log-normal and Gaussian distributions ofthe RSS and TOA measurements by examining the residuals


Fig. 4. Measured wideband path gain (x) as a function of path length. Linearfit (—) is with d = 1 m,n = 2:3, and� = 3:92.

Fig. 5. Q–Q plot of (a)P (dBm) � P (dBm) for RSS data and (b)T � d =c for TOA data compared with a Gaussian quantile.

dBm dBm andvia quantile-quantile plots (normal probability) in Fig. 5. BothRSS and TOA data fit the models well between the2 and

2 quantiles. Using a Kolmogorov–Smirnov (KS) test, wetest the hypothesis versus isnot Gaussian, where is the sample mean of , and isthe sample variance. An identical test is conducted onforthe TOA measurements. For the RSS and TOA residuals, theKS tests yield p-values of 0.09 and 0.50, respectively. In bothcases, we would decide to accept at a level of significanceof .

However, the low p-value for the RSS data indicates thatlog-normal shadowing model in (6) may not fully characterizethe data. In fact, if we use in a two-component Gaussianmixture distribution (with parameters estimated from viathe MLE), the KS test yields a p-value of 0.88. A topic for fu-ture research is to investigate whether the potential benefits ofusing a mixture distribution in the channel model would justifyits additional complexity. The experimental results reported inthe next sections use only the MLE derived under the log-normalshadowing model; these results nevertheless demonstrate goodlocation accuracy.

(a)

(b)

Fig. 6. True (� #T) and estimated ( #E) location using (a) RSS and (b) TOAdata for measured network with four reference devices (X#). Higher errors areindicated by darker text.

B. Location Estimates From Measurements

Four devices near the corners are chosen as reference de-vices. The remaining 40 devices are blindfolded devices. Thefour reference device coordinates and either the RSS or TOAmeasurements or are input to the MLE in (14) or (17).The minimum in each case is found via a conjugate gradientalgorithm. Then, the estimated device locations are comparedwith the actual locations in Fig. 6(a) and (b). To generalizethe results, the RMS location error of all 40 unknown-locationdevices is 2.18 m in the RSS case and 1.23 m in the TOA case.Since shadowing and non-LOS errors are not ergodic, as dis-


cussed in Section IV-A, experimentally determining the MLEvariances would require several measurement campaigns withthe same device geometry but in different office areas. This wasnot possible due to resource and time limitations. Nevertheless,it is instructive to report the CRB for the measured network.We use the measured channel parameters and

ns, the four reference devices used above, and theactual coordinates of all of the devices to calculate the CRBfor given in (12) for and for both TOA andRSS measurements. The quantity is lowerbounded by 0.76 m for the RSS and 0.69 m for the TOA cases.

We also notice that the devices close to the center are locatedmore accurately than the devices on the edges, particularly inthe RSS case. Poor performance at the edges is expected sincedevices have fewer nearby neighbors to benefit their locationestimate.

V. TESTBEDEXPERIMENTATION

To provide an easy means for M2M radio channel measure-ment and location estimation testing, we developed and fabri-cated at Motorola Labs a testbed of 12 prototype peer-to-peerwireless sensor devices with RSS measurement capability. Thedevices have FSK transceivers with a 50-kHz data rate whichoperate in the 900–928 MHz band at one of eight center fre-quencies separated by 4 MHz, which is approximately thecoherence bandwidth of the channel. Devices hop betweencenter frequencies so that RSS measurements can be takenat each center frequency. While one device transmits, otherdevices measure its RSS. Packet transmissions are infrequent,and packets are short; thus, the channel is almost alwayssilent. Devices are asynchronous and use a carrier-sense mul-tiple access (CSMA) protocol. Thus, RX measurements arenot subject to multiuser interference. Every 2 s, each devicecreates a packet of measured RSS data and transmits it toa central “listening” device, which uploads data to a laptopcomputer. The laptop has access to the known coordinates ofthe reference devices and the TX power and the RSS charac-teristic of the devices, as measured prior to deployment. Thelaptop stores the RSS for each pair of devices, each frequency,and each measurement over time.

First, we use the testbed as an easy way to estimate the pathloss exponent . When all of the device locations are known,the laptop uses the path loss versus path length data to estimatethe path loss exponent [25]. After estimating , the blind-folded device coordinates are removed from the laptop, and weoperate the relative location estimation algorithm using the es-timated .

Next, the relative location estimation algorithm averages themeasurements over time (using the most recent four RSS mea-surements), frequency (across eight center frequencies), and thereciprocal channel, resulting in the averaged measurement.The maximum of the MLE in (17) is found using a conjugategradient algorithm, which takes less than 1 s on the Pentiumlaptop. Each second an updated location is calculated and dis-played on a map in a Visual Basic GUI. Real-time tracking ofslow movement (e.g., people walking) is possible.

(a)

(b)

Fig. 7. True (� #T) and estimated ( #E) location for the (a) parking lot and(b) residential home tests using four reference devices (X#). Higher errors areindicated by darker text.

A. Parking Lot Area

Testbed devices are placed in a 9 by 9 m area in a 3-m gridin an empty parking lot area at the Motorola facility. Devicesare kept at a height of 0.35 m. Using the testbed, we estimate

to be 3.2. Then, we place reference devices at the four cor-ners of the area and blindfolded devices at seven of the re-maining 12 spots in the grid (for 11 devices total). Devicesrecord RSS and send packets as described above. The blind-folded devices are then moved to different positions in the gridfor a new trial. Sixteen trials are run. The RMS location errorsfor the individual trials range from 0.9 to 2.4 m. However, bymoving seven blindfolded devices around between positions, werecord enough point-to-point ranges to see what would happenif there were 12 blindfolded devices: one in each spot on thegrid. We use the recorded range data off-line to calculate thatthe RMS error would have been 1.46 m. Furthermore, if we ex-tended the duration of the time averaging from 4 to 32 ranges,we would see the location estimates shown in Fig. 7(a), and theRMS error would reduce to 1.02 m. Since shadow fading is notsevere in this environment, time averaging is effective at im-proving location estimates.


Fig. 8. Map of the grid of sensors in the Perkins home.

B. Residential Home

Next, we test the system in the Perkins home: a single-family,ranch-style house in Sunrise, FL (see Fig. 8). An identical 9by 9 m grid is used in this test, which spanned across manyinterior rooms and an outdoor patio. The obstructions includeindoor walls, furnishings, appliances, and exterior walls andwindows, and is estimated to be 4.0. Here, there are fourreference devices in the corners of the grid and eight otherblindfolded devices. In 16 individual trials, the RMS locationerrors range from 1.0 to 2.7 m. If all device ranges are usedtogether, as described previously, we see the results in Fig. 7(b),in which the RMS error is 2.1 m. This error does not reducesignificantly when the duration of time-averaging is increasedfrom 4 to 32 ranges. Much of the error is due to device #15,which has an error of 4.5 m. As seen in Fig. 8,m, but significant shadowing is caused by the office closetand master bedroom closet that both lie directly in betweenthe two devices dBm dBm , andas a result, the range estimate between the two is 10.5 m.Unfortunately, this shadowing cannot be countered by time orfrequency averaging.

VI. CONCLUSIONS

The motivation for this paper has been to show the accu-racy with which wireless sensor networks can estimate the rel-ative sensor locations. The results should help researchers de-termine if the accuracy possible from relative location estima-tion can meet their application requirements. This paper beganby proving that location estimation variance bounds (CRBs) de-crease as more devices are added to the network. Next, it wasshown that CRBs can be readily calculated for arbitrary num-bers and geometries of devices, and several examples were pre-sented. Sensor location estimation with approximately 1 m RMSerror has been demonstrated using TOA measurements. How-ever, despite the reputation of RSS as a coarse means to esti-mate range, it can nevertheless achieve an accuracy of about 1m RMS in a testbed experiment. Fading outliers can still impairthe RSS relative location system, implying the need for a robustestimator. Future experimentation is needed to verify the vari-ance of location estimators due to the nonergodic nature of shad-owing. Analysis can quantify the effect of “nuisance” channelparameters and can be extended to consider the effects of mul-tiuser interference on sensor location estimation.

APPENDIX

A. CRB for Network Self-Calibration

The diagonal elements of given in (2) are

Since and are independent random variables, and, the expectation of the product is only

nonzero for . Thus, simplifies to the resultin (3). The off-diagonal elements similarly simplify

Here, due to independence and zero mean of the two terms, theexpectation of the product will be zero unless both and

. Thus, the result in (3).

B. Proof of Theorem 1

Compare , which is the FIM for the blindfolded deviceproblem, to , which is the FIM for the blindfolded devicecase. Partition into blocks

where is an matrix, is the scalar Fisher informationfor , and are vectors with th element

Here, we denote the log-likelihood of the observation betweendevices and in (1) as and for the andblindfolded device cases, respectively. Similarly, letand be the joint log-likelihood function in (1) forthe and blindfolded device cases, respectively. Then

Since is a function only of parameters and

.


Thus, diag , where , and. Compare the CRB for

the covariance matrix of the first devices in the anddevice cases, given by and , respectively. Here,

is the upper left submatrix of

where diag

Both and are Hermitian. We know that is positivesemidefinite. Let , be the eigenvalues ofand , be the eigenvalues of the sum,both listed in increasing order; then, if we can show thatispositive semidefinite, then it is known [26] that

(18)

Since the eigenvalues of a matrix inverse are the inverses of theeigenvalues of the matrix

(19)

which proves property 1 of Theorem 1. If in addition, we canshow that tr , then tr tr , and therefore,

. This with (18) implies thatfor at least one . Thus, in addition

to (19)

for some

which implies that tr tr , which provesproperty 2 of Theorem 1.

1) Showing Positive Semidefiniteness and Positive Trace of: The diagonal elements of, are

(20)

If , then and ; thus, .Otherwise, if

Because of reciprocity, the numerator is equal to the square ofthe term in the sum in the denominator. Thus

The equality will hold if is the only member of the set. When condition 2) of Theorem 1 holds, will be strictly

greater than zero. Thus, tr .Next, we show that is diagonally dominant [26], i.e.,

where is given in (20). Since , thus, ,and an equivalent condition is

(21)

If , then , and , and the equalityholds. If , then

Because of condition 1) of Theorem 1

Thus

Since if , we can include in the firstsum all . Since the second sum is , (21) is true.

Diagonal dominance impliesis positive semidefinite, whichproves (19). Note that if includes reference device,the second sum is , and the inequality in (21) is strictly ,which implies positive definiteness of and assures that theCRB will strictly decrease.

C. CRB for Location Estimation

For the elements of , using (7) and (1)

Recall . Thus

Note that ; thus, the log-normaldistribution of RSS measurements meets condition 1) of The-orem 1. The second partials differ based on whether or notand if the partial is taken w.r.t. or . For example

Note that . Thus, the FIM simplifies totake the form in (10). For the TOA case, the derivation is verysimilar, and the details are omitted for brevity.


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Neal Patwari (M’02) received the B.S.E.E. andM.S.E.E. degrees from the Virginia PolytechnicInstitute and State University (Virginia Tech),Blacksburg, in 1997 and 1999, respectively. SinceSeptember 2001, he has been pursuing the Ph.D.degree in electrical engineering at the University ofMichigan, Ann Arbor.

He performed research at the Mobile and PortableRadio Research Group (MPRG) at Virginia Techbetween 1996 and 1999 in radio channel modelingand measurement. He has interned at Comsat Labs,

Clarksburg, MD; the Oak Ridge National Lab, Oak Ridge, TN; and the ArmyResearch Lab, Adelphi, MD. He was a research engineer at Motorola Labs,Plantation, FL between 1999 and 2001, where he conducted research in radiolocation systems and wireless sensor networks. His current research is inestimation and detection in wireless sensor networks. He has published 13technical papers in international journals and conferences and has an issuedpatent.

Mr. Patwari was awarded an NSF Graduate Research Fellowship in 1997.

Alfred O. Hero, III (F’98) was born in Boston, MA,in 1955. He received the B.S. degree in electrical en-gineering (summa cum laude) from Boston Univer-sity in 1980 and the Ph.D. from Princeton University,Princeton, NJ, in 1984, both in electrical engineering.

While at Princeton, he held the G.V.N. LothropFellowship in Engineering. Since 1984, he has beena Professor with the University of Michigan, AnnArbor, where he has appointments in the Departmentof Electrical Engineering and Computer Science,the Department of Biomedical Engineering, and the

Department of Statistics. He has held visiting positions at I3S University ofNice, Sophia-Antipolis, France, in 2001, Ecole Normale Supérieure de Lyon,Lyon, France, in 1999, Ecole Nationale Supérieure des Télécommunications,Paris, France, in 1999, Scientific Research Labs of the Ford Motor Company,Dearborn, MI, in 1993, Ecole Nationale Superieure des Techniques Avancees(ENSTA) and Ecole Superieure d’Electricite, Paris, in 1990, and LincolnLaboratory, Massachusetts Institute of Technology, Cambridge, from 1987 to1989. His research has been supported by NIH, NSF, AFOSR, NSA, ARO,ONR, DARPA, and by private industry in the areas of estimation and detection,statistical communications, bioinformatics, signal processing, and imageprocessing.

Dr. Hero served as Associate Editor for the IEEE TRANSACTIONS ON

INFORMATION THEORY from 1995 to 1998 and again in 1999 and the IEEETRANSACTIONS ON SIGNAL PROCESSING since 2002. He was Chairman ofthe Statistical Signal and Array Processing (SSAP) Technical Committeefrom 1997 to 1998 and Treasurer of the Conference Board of the IEEESignal Processing Society. He was Chairman for Publicity of the 1986 IEEEInternational Symposium on Information Theory (Ann Arbor, MI) and GeneralChairman of the 1995 IEEE International Conference on Acoustics, Speech,and Signal Processing (Detroit, MI). He was co-chair of the 1999 IEEEInformation Theory Workshop on Detection, Estimation, Classification, andFiltering (Santa Fe, NM) and the 1999 IEEE Workshop on Higher OrderStatistics (Caesaria, Israel). He Chaired the 2002 NSF Workshop on Challengesin Pattern Recognition. He co-chaired the 2002 Workshop on Genomic SignalProcessing and Statistics (GENSIPS). He was Vice President (Finance) ofthe IEEE Signal Processing Society from 1999 to 2002. He was Chair ofCommission C (Signals and Systems) of the U.S. National Commission of theInternational Union of Radio Science (URSI) from 1999 to 2002. He has beena member of the Signal Processing Theory and Methods (SPTM) TechnicalCommittee of the IEEE Signal Processing Society since 1999. He is also amember of the SEDD Review Panel of the U.S. National Research Council.He will be President-Elect of the IEEE Signal Processing Society from 2004to 2005. He is a member of Tau Beta Pi, the American Statistical Association(ASA), the Society for Industrial and Applied Mathematics (SIAM), andthe U.S. National Commission (Commission C) of URSI. He received the1998 IEEE Signal Processing Society Meritorious Service Award, the 1998IEEE Signal Processing Society Best Paper Award, and the IEEE ThirdMillenium Medal. In 2002, he was appointed IEEE Signal Processing SocietyDistinguished Lecturer.


Matt Perkins (M’99) received the B.S.E.E. andM.S.E.E. degrees from the University of Florida,Gainesville, in 1996 and 1997, respectively, wherehis research interests included communicationtheory and real-time signal processing related tomagnetic resonance imaging.

Since graduating, he has worked for MotorolaLabs, Florida Communication Research Lab,Plantation, where he has contributed to narrowbanddigital-radio technologies, fluorescent-lamp modu-lation techniques, and indoor and outdoor location

technologies. His current research interests includead hoc wireless sensornetworks and data-fusion techniques.

Neiyer S. Correal (M’99) was born in Bogotá,Colombia. He received the B.S. degree in electronicsengineering from the Pontifical Javeriana University,Bogota, in 1992 and the M.S. degree in electricalengineering from Old Dominion University, Nor-folk, VA, in 1994, where he was involved withspeech recognition at the Speech CommunicationsLaboratory. He received the Ph.D. degree fromVirginia Polytechnic Institute and State University(Virginia Tech), Blacksburg, in 1999.

At Virginia Tech, he was a member of the Mo-bile and Portable Radio Research Group (MPRG). He served as an AssistantProfessor at the University of Florida, Gainesville, and presently is with Mo-torola’s Florida Communications Research Labs, Plantation, FL. His researchinterests include wireless personal area networks,ad hocnetworks, communi-cation theory and real-time signal processing.

Dr. Correal serves as an Associate Editor in the area of Communications Sys-tems Engineering Education for IEEE TRANSACTIONS ONEDUCATION.

Robert J. O’Dea (M’93) received the B.S. degreein electrical engineering from Valparaiso University,Valparaiso, IN, in 1984 and the M.S. and Ph.D.degrees in electrical engineering from NorthwesternUniversity, Evanston, IL, in 1988 and 1992, respec-tively.

From 1987 to 1989, he was a member of theDepartment of Electrical Engineering and ComputerScience at Northwestern University, where he servedas Senior Lab Manager for the Computer ScienceResearch Lab. In 1992, he joined Motorola, where

he is presently Director of Research, Florida Communications Research Lab,Motorola Labs, Plantation, FL. His research interests include digital modu-lation, linear power amplifier efficiency, wireless communication protocols,self-organizing networks, and location/positioning technology. He has sevenissued patents and has authored 13 journal/conference publications.

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