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An RSSI-Based MAP Localization Method withChannel Parameters Estimation in Wireless Sensor

NetworksDaisuke Anzai and Shinsuke Hara

Graduate School of Engineering, Osaka City University3-3-138, Sugimototyou, Sumiyoshi-ku, Osaka 558-8585, Japan

Email: [email protected] and [email protected]

Abstract— This paper proposes a received signal strengthindicator (RSSI)-based maximum a posteriori (MAP) localizationmethod with channel parameters estimation in wireless sensornetworks. The proposed method makes use of not only likelihoodvalue of the location of a target but also a priori knowledgeof the target location. Furthermore, the proposed method alsoestimates channel model parameters with an maximum likelihood(ML) estimation technique, therefore, it can be realized with notroublesome pre-measurement on the channel parameters. Ourtheoretical analyses and experimental results demonstrate thatthe proposed MAP location estimation method is superior to aconventional ML location estimation method in term of locationestimation accuracy.

I. INTRODUCTION

Estimating and tracking the location of a target is one ofthe most important applications in wireless sensor networks.For example, the estimated locations of visitors and customersare used for man-monitoring in exhibitions and supermarkets,and furthermore their estimated locations with time stamps areused for the analysis of their interests.

Several location estimation methods have been so far pro-posed. Generally, time of arrival (TOA)-, time difference ofarrival (TDOA)- angle of arrival (AOA)-based methods showgood location estimation accuracy, however, the three methodsrequire precise synchronization among the local oscillatorsof wireless nodes, several types of signals with differentvelocities and multiple antennas at nodes, respectively [1]–[3]. Therefore, they are disadvantageous in terms of cost andenergy consumption of sensor communication nodes.

On the other hand, RSSI-based location estimation methodis advantageous in terms of cost and energy consumption,because most of the current wireless communication standardshave a function of measuring RSSI in their protocols. Forinstance, the IEEE 802.15.4 standard, which is developed toprovide ultra-low complexity, low-cost, and extremely low-power wireless connectivity among inexpensive devices suchas sensor nodes, has the function in its protocol [4]. Further-more, when the sensing area is full of moving objects in realenvironments, the estimation performances of TOA-, TDOA-and AOA-based estimation methods are worse than an RSSI-based estimation method with ML estimation technique [5].This is because the three methods require a direct wirelesscommunication (LOS: Line-Of-Sight) link to directly measurethe distance for a target, whereas the RSSI-based ML esti-mation method can take into consideration the variation of

measured RSSIs due to multipath fading and shadowing in anNLOS (Non-Light-Of-Sight) link.

In this paper, we propose an RSSI-based MAP localizationmethod with channel parameters estimation in wireless sensornetworks. In order to improve the performance of RSSI-basedML location estimation method, the proposed method makesuse of not only likelihood value of the location of a target butalso a priori knowledge of the target location. In addition, theproposed method estimates channel model parameters with anML estimation technique, therefore, it can be realized withouttroublesome pre-measurement on the channel parameters. Thispaper is organized as follows. Section II shows the conven-tional ML location estimation method. Section III describesthe proposed MAP location estimation method, and SectionVI shows the experimental results for the proposed method.Finally, Section V concludes this paper.

II. ML LOCATION ESTIMATION

A. System Model

In our location estimation system, there are a target nodewhose location is unknown so should be estimated and Nanchor nodes whose locations are known in advance. Theanchor nodes and the target node are put with a fixed height,therefore, we estimate the two-dimensional location of thetarget node instead of its three-dimensional location. The targetnode transmits M packets to the anchor nodes, and eachanchor node measures an RSSI for a received packet. Here, wedefine the locations of the target node and the n-th anchor node(n = 1, 2, · · · , N ) in column vector forms (3×1) respectivelyas

t = [x, y, 0]T (1)

an = [xn, yn, h]T (2)

where h denotes the height between the target node and theanchor nodes.

B. Model of Wireless Communication Link

To accurately estimate the location of the target with RSSI(RSSI means the received signal power in this paper), astatistical model on the RSSI is required, which can well char-acterize the variation of the RSSI in the location estimationarea. From the channel measurement campaigns conducted inrooms, corridors, a shopping street and a foyer of a conferencehall, we came to a conclusion that the RSSI of the IEEE

978-1-4244-2517-4/09/$20.00 ©2009 IEEE 1

802.15.4 signal can be well modeled with the following two-layered model: [6], [7]

P (c) = αr−β (3)

p(P |r) =1

P (c)exp

(− P

P (c)

)(4)

c = [α, β]T (5)

where P , P and r are the received power, the average receivedpower and the distance between a target node and anchornodes, respectively, and p(P |r) is the conditional probabilitydensity function (pdf ) of P when r is given. In (3), c is thechannel parameter vector (2×1), and α and β are the constantsthat are uniquely determined by the location estimation area.

C. ML Location Estimation Method

Defining rn as the distance between the target node and then-th anchor node, it is written as

rn =√

(x − xn)2 + (y − yn)2 + h2. (6)

The unknown parameter vector and the m-th measuredRSSI vector can be written as u = [x, y]T and Pm =[P1m, P2m, · · · , PNm]T , respectively, where Pnm is the them-th RSSI measured at the n-th anchor node. Therefore, thelog-likelihood function on u and c is written as

L(u, c) = log p(P1,P2, · · · ,PM |u, c). (7)

Assuming that Pnm is statistically uncorrelated withPnm′(m �= m′) (temporal whiteness) and Pn′m(n �= n′) (localwhiteness), finally, we obtain

L(u, c) = M

N∑n=1

[log

(1

αrn−β

)−

∑Mm=1 Pnm/M

αrn−β

]. (8)

The ML location/channel parameter estimation gives u and cwhich maximize (8), where (·) denotes the estimate of (·).

III. PROPOSED MAXIMUM A POSTERIORI LOCATION

ESTIMATION

A. Model of A Priori Probability Density Function

In order to introduce MAP estimation into the locationestimation, we derive a priori pdf of the distance between atarget node and anchor nodes. From Bayes’ theorem [8], theposteriori pdf p(r|P ) is proportional to the likelihood functionp(P |r) and the priori pdf p(r), that is,

p(r|P ) ∝ p(P |r) × p(r). (9)

Therefore, combining the priori pdf on r with the conditionalpdf on P leads to the MAP location estimation.

Assume a square location estimation area, where a targetnode is uniformly distributed. In this case, the priori pdf of xand y are written respectively as

p(x) =1X

(−X

2� x � X

2

)(10)

p(y) =1Y

(−Y

2� y � Y

2

). (11)

In (10) and (11), X and Y represent the width and depthof the room, respectively, assuming that the origin of the

Anchor node

Target node

Fig. 1. Location estimation system.

xy plane is located at the center of the room as shown inFig. 1. With (10) and (11), we can derive the priori pdfon r =

√(x − xn)2 + (y − yn)2 + h2. For example, when

xn = yn = a, X = Y = A and h = 0, p(r) results in (12).Fig. 2 shows p(r) for several cases. From these figures,

we can see that the shape of p(r) is complicated but itspeak is exactly located at r =

√(A/2 − xn)2 + h2 or r =√

(A/2 − yn)2 + h2, that is, the minimum distance from theanchor node to the wall of the location estimation area. Sincethe derived priori pdf p(r) is too complicated, it is not suitedfor solving the maximization problem of (9) with it. Therefore,we approximate p(r) with a Rayleigh distribution as follows:

p(r) =

{r

Cσ2 exp(− r2

2σ2

)when h � r

0 otherwise(13)

σ = min[√

(X/2 − |xn|)2 + h2,√

(Y/2 − |yn|)2 + h2]

(14)

where C is a normalized coefficient of the approximated pdfand given by

C =∫ ∞

h

r

σ2exp

(− r2

2σ2

)= exp

(− h2

2σ2

). (15)

Here, we emphasize that the approximated pdf can be deter-mined by the σ given only by (14), namely, the minimumdistance from the anchor node to the wall of the locationestimation area. Fig. 2 includes the simplified priori pdf p(r)given by (13). Note that, for these examples in Fig. 2, thecorrelation coefficients between the simplified and exact pdfsare more than 0.888, namely, the simplified pdf can wellapproximate the exact pdf .

B. Proposed MAP Location/ML Channel Parameters Estima-tion

Defining the measured data vector (M × 1) as

P = [P1,P2, · · · ,PM ]T (16)

the logarithm of the conditional pdf on u when P and c aregiven is written as

log p(u|P, c) ∝ log p(P|u, c) + log p(u)= L(u, c) + log p(u) (17)

where the first term and the second term in the right sideare the likelihood function on u and c given by (8) and the

2

distance ( )

a pr

iori

pdf

of

dist

ance

a priori pdf calculated exactly

approximated pdf

(a) the shape of a room is regular tetragon, and thetarget node and the anchor node are fixed at the sameheight.

distance ( )

a pr

iori

pdf

of

dist

ance a priori pdf calculated exactly

approximated pdf

r

(b) the shape of a room is regular tetragon, and thetarget node and the anchor node are fixed at thedifferent height.

distance ( r )

a pr

iori

pdf

of

dist

ance a priori pdf calculated exactly

approximated pdf

(c) the shape of a room is rectangle, and the targetnode and the anchor node are fixed at the differentheight.

Fig. 2. A priori probability density functions of distance(r).

log-priori probability density function on u given by (13),respectively. From (13), the right side of (17) results in

L(u, c) +N∑

n=1

[log rn − 2 log σ − r2

n

2σ2

]. (18)

In the proposed method, we estimate the target node lo-cation u and the channel parameters c with MAP estima-tion and ML estimation, respectively. Namely, the proposedlocation/channel parameters estimation is divided into thefollowing two stages.

In the first stage, we estimate the target node location uwith MAP estimation. In order to derive an iterative jointlocation/channel parameters estimation, let us introduce theindex for the number of iterations as k for u and c. When ck

has been given, the MAP location estimation gives uk+1 asfollows: (k = 1, 2, · · · )

uk+1 = arg maxu

[L(u, c = ck) + log p(u)] (19)

where arg(·) denotes the argument of (·).On the other hand, in the second stage, we estimate the

channel parameters c with ML estimation using the target nodelocation uk+1 estimated in the first stage. Namely, the MLchannel parameters estimation gives ck+1 which maximizes(17): (k = 1, 2, · · · )

ck+1 = arg maxc

L(u = uk+1, c). (20)

As shown in (19) and (20), we estimate the target nodelocation u and the channel parameters c iteratively in the firstand second stages, so the proposed MAP location estimation

method can be realized with no troublesome pre-measurementon the channel parameters.

Finally, we show the impact of the approximation of thepriori pdf. Fig. 3 shows the root mean square (RMS) locationestimation error for the proposed methods with the exact prioripdf and the approximated pdf by a computer simulation. Fig. 3also shows a result for the ML estimation method. Fromthis figure, the performances of the proposed MAP locationestimation methods are better than that of the ML locationestimation method. Furthermore, the difference between theperformances for the use of the approximated pdf and theexact pdf is little. However, because the approximated pdf isdifferentiable, the proposed MAP estimation can be realizedwith the conjugate gradient algorithm [9], which does notneed so much computational complexity. On the other hand,because the exact pdf is not differentiable, the proposed MAPestimation can be realized only with the greedy algorithm,which needs heavy computational complexity. Therefore, theproposed method with the approximated pdf is advantageousin terms of the computational complexity.

C. Bayesian Cramer-Rao Lower Bound for The ProposedMAP Location Estimation

The quality of the estimate of u can be measured interms of the variance. The Bayesian Cramer-Rao lower bound(BCRLB) [10] provides a lower bound on the minimum errorvariance for MAP estimation. When u are treated as a randomvariable with the known priori probability density functiongiven by (13), the BCRLB is gven by the diagonal elementsof the inverse of the information matrix JT [10]:

JT = JF + JP (21)

p(r) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2πr/A2 if 0 � r � |A/2 − a|πr/A2 + 2r/A2 arcsin

[{2(A/2 − a)2 − r2}/r2]

if |A/2 − a| � r �√

2|A/2 − a|πr/A2 + r/A2 arcsin

[{2(A/2 − a)2 − r2}/r2]

if√

2|A/2 − a| � r � |A/2 + a|2r/A2 arcsin

[{2(A/2 + a)2 − r2}/r2]

+ r/A2 arcsin[{2(A/2 − a)2 − r2}/r2

]if |A/2 + a| � r � |A/2 − a| + |A/2 + a|

r/A2 arcsin[{2(A/2 + a)2 − r2}/r2

]if |A/2 − a| + |A/2 + a| � r �

√2|A/2 + a|

0 otherwise.

(12)

3

Proposed MAP estimation with approximated pdfConventional ML estimation

Proposed MAP estimation with exact pdf

RM

S lo

catio

n es

timat

ion

erro

r [m

]

Number of transmitted packets (M) [packets]

The number of iterations = 20

Computer simulation

Fig. 3. Impact of the approximation of the priori pdf.

where JF and JP denote the Fisher information matrix repre-senting information obtained from the measurements and thepriori information matrix, respectively. In (20), JF is writtenas

JF = −E

{[∂

∂uL(u)

] [∂

∂uL(u)

]T}

. (22)

and the matrix JP is written as [10, ch.2, eq.(290)]

JP = −E

{[∂

∂ulog p(u)

] [∂

∂ulog p(u)

]T}

. (23)

Let I and Iii denote the inverse matrix of JT and its i-th diagonal element, respectively. In this case, the minimumlocation error variance for the proposed MAP estimationσ2

BCRLB is given by

σ2BCRLB = min(var[x] + var[y])

= I11 + I22. (24)

When no priori information is available, that is, JP = O, theMAP estimator becomes equivalent to the ML estimator, soσ2

BCRLB represents σ2CRLB , which is the minimum location

error variance for the ML location estimation.

IV. PERFORMANCE EVALUATION

A. Experimental Setup

In order to evaluate the proposed method, we conductedlocation estimation experiments in Room I and Room II shownin Fig. 4(a) and Fig. 5(a), respectively. Fig. 4(a) shows a sensornode, which is based on the IEEE 802.15.4 standard (2.4GHz-band), and in the experiments, we used the same typeof sensor nodes as a target node and anchor nodes. Fig. 4(b)and Fig. 5(b) show the layout of the anchor nodes in RoomI and Room II, respectively. The heights from the target nodeto the anchor nodes were fixed at 0 m and 1.88 m in RoomI and Room II, respectively. Note that in Room I, the targetnode and the anchor nodes were fixed at the same height andthe shape of the room was rectangle, whereas in Room II, thetarget node and the anchor nodes were fixed at the differentheights and the shape of the room was regular tetragon. Inthe experiments, there were several people walking around inthe rooms, so the direct wireless links between the target nodeand the anchor nodes were frequently shadowed.

(a) Picture.

0.5m

2.9m

6.90

m

4.85 m

2.0m

0.85m

1.5m

2.4m

(b) Layout of anchor node.

Fig. 4. Room I.

(a) Picture.

7.04 m

6.91

m

1.17 m 2.34 m 2.34 m

1.79

m1.

58m

1.79

m

(b) Layout of anchor node.

Fig. 5. Room II.

B. Experimental Results

Fig. 6 and Fig. 7 show the RMS location estimation errorversus the number of transmitted packets M for the proposedMAP location estimation method with the approximated pdfat the number of iterations k = 20 and the initial channelparameter vector c1 = [10−6, 2.0]T in Room I and RoomII, respectively. These figures include the results for the MLlocation estimation method in Room I and Room II, respec-tively, and furthermore, they include the performances by theBCLRB for the proposed MAP estimation and the CRLB forthe ML estimation in Figs. 6 and 7. We can see from thesefigures that the performances of the proposed MAP method forthe experiments and the theoretical bounds were better thanthose of the conventional ML method in both the two rooms.For example, at the number of transmitted packets = 20, ascompared with the conventional ML estimation, the proposedMAP estimation improves the performance by around 30 %and 40 % in Room I and Room II, respectively.

Fig. 8 and Fig. 9 show the RMS location estimation errorversus the number of iterations k with various initial values ofthe channel parameters c1 at the number of transmitted packetsM = 20 in Room I and Room II, respectively. We can seefrom these figures that the proposed MAP location estimationmethod is insensitive to the initial value of the channelparameters and quickly gives a more accurate estimate onthe location of the taget than the conventional ML estimationmethod.

4

RM

S lo

catio

n es

timat

ion

erro

r [m

]


BCRLB for the proposed MAP estimation

CRLB for the conventional ML estimation


Proposed MAP estimationConventional ML estimation

Experiment

Fig. 6. RMS location estimation versus the number of transmitted pack-tes (M ) in Room I.

RM

S lo

catio

n es

timat

ion

erro

r [m

]


Proposed MAP estimationConventional ML estimation

Experiment




Fig. 7. RMS location estimation versus the number of transmitted pack-tes (M ) in Room II.

V. CONCLUSIONS

In this paper, we have proposed a MAP location estimationmethod with the ML channel parameters estimation. We haveshown that, in the experiments, the proposed MAP estimationmethod improves the location estimation accuracy by 30 % and40 % in Room I and Room II, respectively, as compared witha conventional ML location estimation method. Furthermore,the proposed MAP estimation method is insensitive to theinitial value of the channel parameters and quickly gives amore accurate estimate on the location of the target than theconventional ML estimation method. The pdf approximationwith a Rayleigh distribution is uniquely determined only bythe minimum distance from the anchor node to the wall of thelocation estimation area, so the proposed method is applicablefor location estimation areas with any shapes.

ACKNOWLEDGEMENT

This study was supported in part by a Grant–in–Aid forScientific Research (No. 19360177) from the Ministry ofEducation, Science, Sport and Culture of Japan.

Number of iterations

RM

S lo

catio

n es

timat

ion

erro

r [m

] Experiment

the conventional ML estimation

the proposed MAP estimation



The number of transmitted packets = 20

Fig. 8. RMS location estimation error versus the number of iterations (k)in Room I.

Number of iterations

RM

S lo

catio

n es

timat

ion

erro

r [m

] Experiment

the proposed MAP estimation

the conventional ML estimation



The number of transmitted packets = 20

Fig. 9. RMS estimation error versus the number of iterations (k) in RoomII.

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