78 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 3, NO. 1, FEBRUARY 2014

New Decentralized Positioning Schemes for Wireless Sensor NetworksBased on Recursive Least-Squares Optimization

Chin-Liang Wang, Fellow, IEEE, Dong-Shing Wu, Member, IEEE, and Kai-Jie Yang, Member, IEEE

Abstract—This letter presents a recursive least-squares (RLS)optimization process to solve the positioning problem for wirelesssensor networks, where a recursive-in-time cost function isfirst defined and then an iterative decentralized algorithm isderived. It is shown that the RLS scheme is equivalent to theincremental subgradient method with an appropriate variablestep size for each iteration. With this, we further replace eachgradient value by its “sign” to form a reduced-complexity RLS(RCRLS) scheme. Simulation results show that RCRLS has someperformance degradation as compared to RLS, but both of themoutperform previous related methods.

Index Terms—Decentralized methods, positioning, recursiveleast-squares optimization, wireless sensor networks.

I. INTRODUCTION

LOCATION estimation or positioning of a target inwireless sensor networks can be categorized into two

types, centralized and decentralized methods [1]–[5]. For acentralized approach, local measurements are passed overmultiple hops to a fusion center for estimating the targetlocation. In contrast, location estimation for a decentralizedapproach is carried out by a distributed optimization processthat is sequentially performed by a set of sensor nodes, each(except the first one) making a location estimate based onits own observation and the latest location update from theprevious sensor node. According to the results given in [6] and[7], a decentralized positioning scheme generally has betterenergy efficiency than a centralized one for large-scale densenetworks.

Recently, several decentralized positioning methods usingreceived signal strength (RSS) measurements have been pro-posed for wireless sensor networks in the literature [7]–[12].By relating a least-squares (LS) location estimation problemto a convex feasibility problem, a projection onto convex sets(POCS) method was investigated in [8] and [9]; to attainbetter positioning accuracy, a projection onto the nearest localminimum (PONLM) scheme was further developed in [10].Although closed-form expressions are given for the projectionphase of POCS-based schemes, the orthogonal projectionoperation would involve high computational complexity. Withthe growing demands of lower computational complexity and

Manuscript received October 17, 2013. The associate editor coordinatingthe review of this letter and approving it for publication was A. Conti.

This work was supported by the National Science Council of the Republicof China under Grant NSC 100-2221-E-007-059-MY3.

C.-L. Wang is with the Department of Electrical Engineering and theInstitute of Communications Engineering, National Tsing Hua University,Hsinchu, Taiwan 30013, Republic of China (e-mail: clwang@ee.nthu.edu.tw).

D.-S. Wu and K.-J. Yang were with the Institute of Communications En-gineering, National Tsing Hua University, Hsinchu, Taiwan 30013, Republicof China (e-mail: {d929603, d919608}@oz.nthu.edu.tw).

Digital Object Identifier 10.1109/WCL.2013.111713.130734

better positioning accuracy, a normalized incremental sub-gradient algorithm and another similar one called “reduced-complexity decentralized positioning (RCDP) method” wereproposed in [11] and [12], respectively, to solve the LS lo-cation estimation problem; however, no analytical procedureswere given for the selection of an appropriate step size forthese two algorithms to ensure good performance.

In this letter, we present a new decentralized positioningmethod based on recursive least-squares (RLS) optimizationfor wireless sensor networks, where a recursive-in-time costfunction is first defined and then an iterative algorithm isderived to approach the optimal solution. During each iterationof the RLS scheme, a sensor node computes a current estimateof the target location by taking a weighted average of theprevious local estimate (i.e., the estimate obtained at theprevious iteration) and its projection point on a circle centeredat the sensor node, and then the sensor node broadcasts thisnew local estimate for the next iteration. It is shown thatthe RLS scheme is equivalent to the incremental subgradient(IG) method [13] with an appropriate variable step size foreach iteration, which ensures convergence to a stationarypoint. To simplify the RLS method, we further replace thegradient value in each dimension with its “sign” (i.e., +1 or−1) to form a reduced-complexity RLS (RCRLS) scheme. Itis noteworthy that, unlike the related algorithms in [6]–[13]which might involve a complicated process to predeterminethe step size, both of the RLS and RCRLS methods adopt ansimple effective rule for updating the step size at each iteration.Simulation results show that the RCRLS scheme has somedegradation in positioning accuracy as compared to the RLSmethod, but both of them gain performance improvementsover the previous RCDP, POCS, and PONLM schemes.

II. BACKGROUND

Consider a network with uniformly deployed sensor nodesin a two-dimensional area, where the ith sensor node is ata known location pi ∈ �2. Assume that there is a targetbroadcasting a signal with power PT and that N active sensornodes receive the signal successfully to participate in thepositioning process. Based on the RSS measurement at theith sensor node, denoted as PR,i, the path loss from the targetto the ith sensor is measured as Li = PT − PR,i. Referringto [14], the path loss measurement Li is described by a log-distance propagation loss model given as

Li = PL0 + 10n log10(di/d0) +XiΔ= PL(di) +Xi (1)

where PL(di) = PL0 + 10n log10(di/d0) (in dB) is theideal path loss corresponding to the Euclidean distance di =‖θ∗ − pi‖ from the target location θ∗ ∈ �2 to the ith sensor

2162-2337/14$31.00 c© 2014 IEEE

WANG et al.: NEW DECENTRALIZED POSITIONING SCHEMES FOR WIRELESS SENSOR NETWORKS BASED ON RECURSIVE LEAST-SQUARES . . . 79

node, PL0 (in dB) is the reference path loss at a positionwith distance d0 from θ∗, n is the path loss exponent forthe environments, and Xi (in dB) is a zero-mean Gaussiandistributed random variable with variance σ2

X , representingthe log-normal shadowing effect.

As described in [7]–[12], the target location estimation canbe formulated as a LS optimization problem as follows:

θLS = arg minθ∈�2

N∑i=1

(Li − PL(‖θ − pi‖))2

Δ= arg min

θ∈�2

N∑i=1

fi(θ)

(2)

where the objective function is a sum of N local cost functionsgiven by fi(θ) = (Li−PL(‖θ − pi‖))2, i = 1, ..., N , with theith one representing the squared error between the measuredand the ideal path loss from location θ to sensor node i.

III. NEW POSITIONING SCHEMES BASED ON RECURSIVE

LEAST-SQUARES OPTIMIZATION

A. The Proposed RLS Positioning Scheme

To solve (2) in an iterative decentralized manner, we fol-low the concept of the well-known recursive least-squaresalgorithm [15] to define a recursive-in-time cost functionFk(θ) =

∑ki=1 fi(θ) and then to modify the location estima-

tion problem as follows:

θk = arg minθ∈�2

Fk(θ)

Δ= arg min

θ∈�2

k∑i=1

fi(θ), k = 1, 2, 3, ..., K(3)

where θk is computed at the kth iteration and K is the totalnumber of iterations used. Specifically, θ1 is generated atiteration 1 by minimizing f1(θ), θ2 is generated at iteration 2by minimizing f1(θ)+f2(θ), and so on. Due to the recursivenature, (3) is referred to as an RLS optimization process. Notethat the RLS optimization process maintains a solution that isoptimal with respect to (3) at each iteration, and the solutionθK is equal to θLS in (2) theoretically if K = N . However, arealizable iterative process would not be able to yield the θLS

solution. To make θK approach θLS as much as possible,we can set K > N and let all the active N sensor nodesbe cyclically reused until K iterations are finished. For easeof presentation, we denote the extended ordered sequence ofsensor nodes operating in the recursive positioning process as{1, 2, . . . , k, k + 1, . . . , K}, where sensor node k mod Nacts at the kth iteration.

Taking ∂Fk(θ)/∂θ| θ=θk= 0, we have

∂Fk(θ)

∂θ

∣∣∣∣θ=θk

=20n

ln 10

k∑i=1

PL(‖θ − pi‖)‖θ − pi‖2

{θ − [pi +Li(θ − pi)

PL(‖θ − pi‖)]}∣∣∣∣∣θ=θk

Δ=

20n

ln 10

k∑i=1

Γi(θ)(θ − PCi(θ))

∣∣∣∣∣θ=θk

= 0

(4)

where Γi(θ)Δ= PL(‖θ − pi‖)/ ‖θ − pi‖2 and PCi(θ)

Δ= pi+

Li(θ−pi)/PL(‖θ − pi‖), which can be interpreted as the pro-jection point of θ onto the circle Ci = {y ∈ �2 : ‖y − pi‖ =Li ‖θ − pi‖ /PL(‖θ − pi‖)}. It is difficult to solve (4) due tothe nonlinearity of {PCi(θ)}ki=1 and {Γi(θ)}ki=1. Consideringthe limited computational capability of each sensor node,we aim to implement the RLS optimization process in (4)for each iteration in a recursive and decentralized manner.With an initial estimate θ0, the location estimate θ1 at sensornode 1 for the first iteration can be approximately obtainedfrom (4) by substituting θ0 for θ in PC1(θ) and Γ1(θ),i.e., θ1 = Γ1(θ0)PC1(θ0)/Γ1(θ0) = PC1(θ0). Similarly,using θ = θ1 for computing PC2(θ) and Γ2(θ) in sensornode 2 at the second iteration, the local estimate becomesθ2 =

∑2i=1 Γi(θi−1)PCi(θi−1)/

∑2i=1 Γi(θi−1). Following

this procedure, we can express the local estimates at the(k − 1)th and kth iterations (θk−1 and θk) as follows:

θk−1 =

k−1∑i=1

Γi(θi−1)PCi(θi−1)/

k−1∑i=1

Γi(θi−1) (5)

θk =

k∑i=1

Γi(θi−1)PCi(θi−1)/

k∑i=1

Γi(θi−1). (6)

Rewriting (6) as

θk =

(k−1∑i=1

Γi(θi−1)PCi(θi−1)/k−1∑i=1

Γi(θi−1))k−1∑i=1

Γi(θi−1)

k∑i=1

Γi(θi−1)

+Γk(θk−1)PCk

(θk−1)k∑

i=1

Γi(θi−1)

(7)and substituting (5) into the brackets of (7) yields the follow-ing recursive relation between θk−1 and θk:

θk =

θk−1

k−1∑i=1

Γi(θi−1)

k∑i=1

Γi(θi−1)

+Γk(θk−1)PCk

(θk−1)k∑

i=1

Γi(θi−1)

= (1 − Γk(θk−1)

D(k))θk−1 +

Γk(θk−1)

D(k)PCk

(θk−1)

(8)

where D(k) Δ=

∑ki=1 Γi(θi−1), k = 1, 2, ...,K , can be

regarded as the accumulated reliability information up toiteration k. Given an initial value D(0) = 0, D(k) can alsobe determined recursively by

D(k) = D(k−1) + Γk(θk−1). (9)

In summary, the proposed RLS positioning scheme can beformed as follows:

1) k = 0: Use the location of sensor node 1 p1 as the initiallocation estimate θ0 and set D(0) = 0.

2) k = 1, 2, ...,K: Compute θk based on (8) and (9) insensor node k mod N at the kth iteration.Fig. 1 shows how the proposed algorithm is realized in adecentralized manner among the active sensor nodes. During

80 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 3, NO. 1, FEBRUARY 2014

1LSensor Node1

1 modk NLmodk NL

modK NL

(1)1, D

Sensor Nodek–1 mod N

Sensor Nodek mod N

Sensor NodeK mod N

( 2)2

ˆ , kk D

( 1)1

ˆ , kk D

( )ˆ , kk D

( 1)1

ˆ , KK D

Final EstimateˆK 0

Initial Estimateˆ

Fig. 1. Decentralized realization of the proposed RLS positioning algorithm.

the kth iteration, sensor node k mod N computes Γk(θk−1),PCk

(θk−1), D(k), and θk according to its path loss measure-ment Lk mod N and the location estimate θk−1 along withthe accumulated reliability information D(k−1) passed fromsensor node k − 1 mod N . Then it transmits the results tosensor node k+1 mod N for further processing at the (k+1)thiteration.

It is easy to check that the gradient of the kth local costfunction defined in Section II can be expressed by

∇fk(θ) =20n

ln 10(PL(‖θ − pk‖)− Lk

‖θ − pk‖2)(θ − pk)

=20n

ln 10(Γk(θ)θ − Γk(θ)PCk

(θ)) .

(10)

Substituting (10) into (8), we obtain an alternative recursiveform as follows:

θk = θk−1 − αk · ∇fk(θk−1) (11)

where αk = [ln 10/(20n)]/D(k) is a variable step size mono-tonically decreasing with k. It can be seen that the proposedRLS scheme has the same iterative form as the IG method[13]; the main difference is that the former adopts a simplerecursive procedure to update the step size at each iteration,but the latter involves a complicated process to predeterminethe step size appropriately. Note that D(k) Δ

=∑k

i=1 Γi(θi−1),where Γi(θi−1) = PL(||θi−1 − pi||)/||θi−1 − pi||2 is thenormalized ideal path loss from the target location esti-mate θi−1 to sensor node pi. Since Γi(θi−1) is boundedas γmin < Γi(θi−1) < γmax, limk→∞ D(k) = ∞ andlimK→∞

∑Kk=1 (1/D

(k)) = ∞. Thus, we have limk→∞ αk =0 and

∑∞k=0 αk = ∞. This means that αk satisfies the

convergence condition of the IG method given in [13], and theproposed RLS positioning scheme can converge to a stationarypoint if all the active N sensor nodes are cyclically reusedenough times such that the total number of iterations K ismuch greater than N . It should also be mentioned that theRLS optimization problem given in (3) is nonconvex, so thestationary point may be a local minimum, but not the globalminimum.

B. The Proposed Reduced-Complexity RLS (RCRLS) Position-ing Scheme

It can be seen from (10) and (11) that the proposed RLSmethod involves gradient operations. For simplification, wefurther propose a reduced-complexity RLS (RCRLS) schemein which each gradient value is replaced with its “sign” (i.e.,+1 or −1) as follows:

θk = θk−1−αk · [sgn(∂fk(θk−1)

∂θ(1)k−1

), sgn(∂fk(θk−1)

∂θ(2)k−1

)]T (12)

where θk = [θ(1)k , θ

(2)k ]T . Letting pk = [p

(1)k , p

(2)k ]T , we have

sgn(∂fk(θk−1)/∂θ(i)k−1)

= sgn(PL(||θk−1 − pk||)− Lk) · sgn(θ(i)k−1 − p(i)k ), i= 1, 2.

(13)As compared to the RLS scheme, the RCRLS scheme reducesfour multiplication operations and one division operation foreach iteration. It could be regarded as an improved version ofthe normalized incremental subgradient algorithm in [11] andthe RCDP method in [12], where an appropriate variable stepsize is employed instead of a predetermined fixed step size.Similar to the related algorithms in [11] and [12] which canconverge to a stationary point under the condition that the stepsize αk is in the range (0, 2), the convergence of the proposedRCRLS algorithm with αk = [ln 10/(20n)]/D(k) ≤ 1 isguaranteed. As compared to previous decentralized approachesin [7]–[12] in terms of communication requirements, the pro-posed RLS and RCRLS schemes need additional transmissionof D(k) to update the step size at each iteration.

IV. COMPUTER SIMULATION RESULTS

In order to verify the performances of the proposed RLSand RCRLS methods, a number of computer simulations wereperformed on a network with 6 × 6 sensor nodes uniformlydistributed in a 50 × 50 meter2 area. A target was randomlylocated with a uniform distribution in the area for 10, 000trials, and it broadcasted a signal to all sensor nodes in eachtrial. The path loss measurements at the sensor nodes weremodeled according to a retail store environment [14] withPL0 = 34.7 dB, n = 2.2, and σ2

X = 75.69 for (1). Thesensor nodes switch to join the positioning procedure onlywhen their path loss measurements are smaller than a thresholdof 63 dB; this arrangement results in that the average numberof active sensor nodes N is around 12. The ordered sequenceof active sensor nodes operating for the iterative positioningprocess was in a counterclockwise manner, where the sensornode with the smallest path loss measurement (i.e., with thelargest RSS measurement) was selected as the first one andall the active sensor nodes were cyclically reused if the totalnumber of iterations (K) was greater than N . For the purposeof comparison, the POCS, PONLM, RCDP, and IG methodswere also simulated on the same environment and their stepsizes were selected based on the criteria in [8] and [10].

Fig. 2 presents the root mean-squared error (RMSE) of thelocation estimate versus the number of iterations for the sixschemes compared, where RMSE

Δ=

√E[||θ − θ∗||2]. It is

clear that the proposed two approaches have superiority over

WANG et al.: NEW DECENTRALIZED POSITIONING SCHEMES FOR WIRELESS SENSOR NETWORKS BASED ON RECURSIVE LEAST-SQUARES . . . 81

20 40 60 80 100 120 140

2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

Number of Iterations

RM

SE (m

)

POCSPONLMRCDPIGRCRLSRLS

Fig. 2. The RMSE versus the number of iterations for different decentralizedpositioning methods.

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance Estimation Error (m)

CDF

POCSPONLMRCDPIGRCRLSRLS

Fig. 3. The CDF of the distance estimation error for different decentralizedpositioning methods with K = 24.

the others in terms of the converged RMSE, where the RLSalgorithm is the best one of them. Fig. 3 shows simulationresults of the cumulative distribution function (CDF) of thedistance estimation error for these methods with K = 24.We can see that the distance estimation error is within 4.235,4.69, 4.875, 4.895, 5.055, and 5.165 meters among 90% ofthe trials for the RLS, RCRLS, IG, RCDP, PONLM, andPOCS methods, respectively, where the proposed RLS andRCRLS approaches have better positioning accuracy thanthe other four. It is worth mentioning that the larger thenumber of iterations K , the more performance improvementsthe proposed schemes will be. The reason why the proposedalgorithm outperforms the others is that the step size is updatedat each iteration based on local observations and the reliabilityinformation accumulated so far, rather than remains unchangedthroughout the iteration process. In contrast to RLS, RCRLSreduces the computational complexity with some performancedegradation, showing a tradeoff between performance andcomplexity.

V. CONCLUSION

In this letter, we have presented a new decentralized RLSpositioning method for wireless sensor networks through theminimization of a recursive-in-time cost function. The RLSscheme can be realized in an iterative distributive manner,where a location estimate is computed at the current sensornode by taking an appropriately weighted average of the pre-vious local estimate obtained at the previous sensor node and

its projection point on a circle centered at the current sensornode. It has been shown that the proposed iterative algorithmis equivalent to the IG method with an appropriate variablestep size for each iteration, which ensures convergence to astationary point. To simplify the RLS method, we have furtherproposed an RCRLS scheme with each gradient value replacedby its “sign”. Simulation results have been given to show thatRCRLS has some performance degradation as compared toRLS, but both of them have better positioning accuracy thanthe previous related schemes.

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