Estimating Distances via Received Signal Strength and Connectivity in Wireless Sensor Networks

Noname manuscript No. (will be inserted by the editor) Estimating Dis tances via Receiv ed S ignal S tr ength and Connectiv ity i n Wir eless Sensor Networks Qing Miao, Baoqi Huang ∗ and Bing Jia Recei ve d: date / Accepted: date Abstract Distance estimation is vital for localization and many other applications in wireless sen so r networks (WSNs). Particularly , it is desirable to imp lement distance estimation as well as localization with out u sin g specific hardware in low-cost WSNs. As such, both the r eceiv ed signal strength (RSS) b ased approach an d the connec tivity based approa c h have gained much attention. The RSS based approach is suitable for estimatin g short distances, wh e reas the con nec- ti vity based appro ach obtains relatively good perfor mance for estimating long distances. Considering the complemen- tary featu res of these two approache s, we propose a f u sion method based on the max imum-likelihoo d estimator (MLE) to estima te the distance between any pair of neighb oring nodes in a WSN th r ough efficiently fusing the inf ormation from the RSS and local connectivity . Additionally , the method is rep orted u nder the prac tical log-norm al shad owing mod el, and the associated Cra mer-Rao lower bound (CRLB) is also derived for perfor m ance analysis. Both simula tio ns a n d ex- periments based o n pr actical m easurements are carried out, and demo nstrate th at the pr oposed m ethod outper f orms any single ap proach an d app roaches to the CRLB as well. Keywords Distance Estimatio n, Maximum-Likeliho od Estimator, Erro r Distributions, Cramer-Rao lower bo und 1 INTR O DUCTION W ireless sensor networks (WSNs), comp o sed of hun dreds or thousands o f sma ll and inexpensive nodes with constrained computin g power , limited me m ories, an d short b attery life- time, can be used to mo nitor and co llect d ata in a r egion of interests. Accurate and lo w- cost n ode localization is im- Inner Mongolia Univ ersity , Hohhot, 010021, China, E-mail: 13734812 507@163.com, cshbq@imu.edu.cn and ji- abing@imu.edu.cn portant fo r various application s in WSNs, and thus, gr e at efforts have b een devoted to developing various localization algorithm s, categor ized into distance based algorithms a nd connectivity-based algorithm s [1]. The d istance-based local- ization algo rithms rely on d istance estimates an d are able to achieve relatively go o d localization accuracy , whereas the connectivity-based localization algorithms g enerally achieve coarse-gr a ined localization ac curacy since only local con- nectivity informa tio n (the nu mbers of the comm on an d n on- common one-ho p neighb ors) is e m ployed f or distance esti- mation. Besides, distance estimation is a lso useful for sen- sor network ma n agement, such a s topolog y con trol [2, 3] and bound ary d e tection [4, 5]. In reality , distance estimation ca n be realized by using informa tio n such as RSS, time of arriv al (TO A) and time difference of arri val (TDOA) [1]. The RSS app r oach (using RSS measuremen ts) does not require a ny de d icated hard - ware, but is ab le to provid e coarse-g rained distance esti- mates; in c ontrast, th e TOA and TDOA m ethods can pr o - vide distan ce estimates with h igh accuracy at the cost of extra ha rdware, but it is unaffordab le to equip each sensor with a dedicated measurem ent d evice in a large-scale WSN due to the costs in both hardware and energy . Therefo re, it is of great impo rtance to enhance the accuracy of low-cost distance mea su rement approach es, and many efforts have been impo sed in the literature. Due to its intrinsic simplic- ity and in depend ence of ded icated ha r dware, th e RSS-based distance metho ds h ave gained m uch attraction [6 – 9]. But, both simulations an d theo retical ana ly sis ind ica te that the perfor mance of the RSS-based meth o ds is relatively po or , and d egrades with the increasing actual distances [6, 7, 10]. Hence, it is n ecessary to advance the RSS-b a sed metho ds by adopting various tech n iques [11, 12]. Apart fr om RSS measuremen ts, local co nnectivity inf o r- mation o f e ach no de is also ind ependen t of extra device, and can be em ployed to estimate distan c e s from itself to other 2 Qing Miao, Baoqi Huang ∗ and Bing Jia neighbo ring n odes [10, 13, 14]. However , Un like the RSS- based method s that retu rn ideal estimates f or short distances, the connectivity-based methods o btain relatively good per- forman ce for estimating lon g d istances. Therefor e, the com- plementary features of these two typ es of distance estima- tion meth ods mo tivate us to design a fusion m e thod which is able to sufficiently exploit the advantages o f both typ es of methods. In this paper, th e maximum -likelihood estimator (MLE) is utilized to efficiently fuse th e inf ormation from the RSS measuremen t and local conne cti vity , so as to provid e good perfor mance r egar d less the actual sizes of distances to b e estimated. The streng th s of the pro p osed fusion method lie in the fo llowing aspects. Firstly , it is well kn own that the MLE is asymp totically optimal, indicating th a t the pro posed method is able to obtain superior perform a nce. Secondly , the practical log- normal (shadowing) m o del is a d opted to ch ar- acterize the probab ilistic distributions of the erro rs re spec- ti vely ind uced by the RSS-based m e th od an d th e co nnectivity- based method , so as to ensu r e th e ap plicability of the pro- posed m ethod. Thirdly , the Cramer-Rao lower bou nd (CRLB) associated with the pr oposed metho d is also formu lated and can be used to ev aluate the optimality of the propo sed method. Finally , both exper iments based o n measur e ments in a real en v ironmen t and simulatio ns are carried out to thoroug hly validate the effecti vene ss o f the p roposed method . Howe ver, ev e n if the p roposed m ethod o utperfo rms the o ther two a vail- able method s and a p proach es to the CRLB in mo st time, it still suffers from a few limitations: it is compu tationally ex- pensive due to th e co mplicated cost function inv olved; as the con nectivity-based method , the perf ormance also relies on the sensor density . This study no t on ly contr ibutes to improving senso r localization in low-cost WSNs, but also paves the way for a dvancing many other researches and ap- plications relying o n inter-node d istances. The rema in der o f th e paper is organized as follows. Sec- tion 2 b riefly revie ws related works in the literature. Section 3 introdu ces the WSN m odel and both the RSS-based an d connectivity-based distance e stimation methods. Sec tio n 4 presents the models of both the RSS-based and conn ectivity- based methods, and proposes th e fusion method based o n th e MLE and for mulates the correspo nding CRLB. Section 5 re - ports the pe rforman ce o f the pro posed method throug h both simulations and experiments. Finally , section 6 co n cludes the p aper and sheds lights on f uture works. 2 RELA TED WORKS In this sectio n, we sh a ll brie fly review the stud ie s on distance estimation in low-cost WSNs, wh ich can be categorized into RSS-based m ethods and con nectivity-based meth ods. The RSS-based me th ods infer distance s from power losses incurred by signals travelling between transmitter and r e- ceiv e r as long as afte r the model dep icting the relatio nship between p ower losses and d istan ces is available. In [6], an estimator was designed based o n th e ML E an d th e log -norm al model, but the theoretical analy sis ind ica ted that the esti- mator is inefficient in the sense that th e er ror variance in- creases expo nentially with p owers. However , th e p r actice in [7, 9] reveal that the RSS-based distance estima tio n is un - reliable. Additio n ally , th e distance estimation is e ven com - plicated in in door environments since th e factors, like fur- niture, h and grip and human bodies, affect th e distance esti- mation [ 8]. Hence, in [11] a dy namic calibr ation me th od was propo sed to update th e log- normal mod el par ameters wh ich fluctuate with environmen ta l chang es, and in [12], an av- eraging me th od based on m u ltichannel RSS measureme n ts was pr esented to mitigate the variability of RSS measure- ments. Th erefore, it can b e con cluded that it is still chal- lenging to app ly the RSS-based m ethods. The co n nectivity-based methods infer distances from lo- cal co nnectivity information am ong d ifferent nodes in WSNs [10, 15, 16]. The n eighbor hood inter section distance estima- tion scheme (NIDES) presented in [15] he u ristically relates the distance, e.g. fr om node A to node B, to an easily ob- served ratio, i.e. the number o f th eir co mmon imme diate neighbo rs to the numbe r of immediate neighb ors of A, an d then perf o rms distance estimation at n ode A acc ording to this ratio and other a priori kn own informatio n. NIDE S as- sumes a unit disk model, n amely th at the comm u nication coverage o f e ach no de is a perfect d isk , and all nodes ar e unifor m ly and rando mly deployed in the WSN. Its enha n ced version presented in [16] adapted the ratio b y taking in to ac - count the number o f immediate n eighbo r s of node B, and heuristically stated that NI DES cou ld be applied un der arbi- trary co mmunica tio n m odels. In [10], a novel m e th od is pre- sented ba sed on the MLE u n der a g eneric chann el mod e l, including the unit d isk mod e l an d the more re a listic log- normal mod el, and its error ch aracteristics were analyzed in light of the CRLB. Howe ver , the p erform a n ce of the co n nectivity- based metho d obtains obvious erro rs when e stimating short distances. In summar y , both of the above meth ods a r e restricted in practical ap plications, but fo rtunately , are com plementary to each oth er , which motiv ates u s to combine them to obtain better perfo rmance. As such, this paper p resents a fu sion method to estimate distances by makin g use o f RSS mea- surements and local c onnectivity u nder the p ractical log- normal m odel. 3 PRELIMIN ARIES This sectio n first briefly introduc e s the static WSN model which is con sidered in this p a per , an d then elabo rates th e RSS-based and conn ectivity-based methods, respec tively . Through - out th is pap er , we shall use the fo llowing ma thematical n o- Estimating Distances via Receiv ed Si gnal Strength and Connecti vi t y in Wire l ess Sensor Networks 3 tation: p ( · ) d enotes th e p r obability d e nsity fun ction of an ev e n t, and E( · ) d enotes the statistical expectation. 3.1 The WSN Mod el In a static WSN, no d es are often assum ed to be rand omly and u n iformly distributed on accoun t of the random nature of network d eployment, e.g. n odes being dropped f rom a fly- ing plan e. Since a ho mogene o us Poisson process provid es an accurate model fo r the unifor m distribution of nod es as the n etwork size app roaches infinity , we d efine th e static WSN to be deployed over an infin ite p lane accord ing to the homog eneous Po isson pr ocess of intensity λ . 3.2 The RSS-b a sed Metho d The RSS-based metho d estimates the distance between any pair of n odes using the received signal power, i.e. RSS. When a sign al is prop a g ated between tr ansmitter an d receiver , the power loss or attenu ation is un av oidab le, and genera lly rises with in creasing the separation b etween tran smitter and re- ceiv e r . Moreover, as is common ly mad e in both theoreti- cal studies ( e.g. [17 – 19]) and experim ental studies ( e . g. [20, 21]), the power loss can b e formulated by usin g th e log- normal m odel, na mely P R ( d ) (dBm) = P R ( d 0 ) (dBm) − 10 α log 10 d d 0 + Z , (1) where P R ( d ) (dBm) is the rec e ived signal power at d in dBm, P R ( d 0 ) (dBm) is the mean r eceiv e d signal power at a refer- ence distance d 0 in dBm, α is the path loss exponen t, and Z is a rand om variable representing th e shadowing effect, normally distributed with m e a n zer o and variance σ 2 dB . Based on the log -norma l mo del in (1), it is stra ig htfor- ward to infer the d istance d f r om th e r eceiv e d signal power P R ( d ) b y using any p arameter estimator . For instance, ˆ d R is defined to be the d istan ce estimate between two nod es via an associated RSS mea su rement, and can be fo rmulated as follows (see [2 2] fo r mo r e details) ˆ d R = 10 P R ( d 0 )( dBm ) − P R ( d )( dBm ) 10 α d 0 . (2) 3.3 The Con nectivity-based Method The co nnectivity-based m ethod estimates the distance be- tween any p a ir o f n eighbor ing nodes on the basis of their lo- cal connectivity inform a tion. In this subsection , we present this me th od und e r b oth the simple unit disk model and the generic chann el mod el. Specifically , the unit disk mo del as- sumes an idea l co m munication coverage fo r each nod e , i.e. Fig. 1 The communication cov erage of two nodes under the unit disk model. a perfe ct disk with the rad ius of r , whereas the generic ch a n- nel model, including the log-norm al model, takes into con- sideration the rando m noises (e.g. th e shad owing effect) in the commu nication chann els, so as to characterize the com - munication coverage in a more p ractical way . 3.3.1 The Unit Disk Model Case Giv en a static WSN, suppo se two nodes A and B with co - ordinates ( x A , y A ) , ( x B , y B ) and separation d ( d ≤ r ) an d the disks with the common rad ius r represent their in divid- ual communicatio n coverage und er the unit d isk mo del, as shown in Fig. 1. Because of d ≤ r , the two disks intersect and cr eate th ree disjoin t regions. Regarding r as constant, define S = πr 2 and f ( d ) to be the are a of the midd le region in Figu re 1, where f ( d ) = 2 S π arccos  d 2 r  − d r r 2 − d 2 4 . (3) It is obvious tha t the node s residing in the m iddle region are comm on immediate n eighbor s of A and B, the nod e s re- siding in the left (or right) one are non-comm on immed iate neighbo rs of A (or B). De fine th ree rand om variables M , P , and Q to be the numb ers o f the three categories of ne ig h- bors; acc ording to the assumption of the Poisson point p r o- cess, they are mutually ind ependen t and Poisson with means λf ( d ) , λ ( S − f ( d )) and λ ( S − f ( d )) , a s p ointed out in [23]. Howe ver, the a ctual values of M , P , and Q can be easily obtained after A and B exchange their neighbo rhood info r- mation. On th e b asis o f th e o bservations of M , P and Q and the method of MLE, the distance estimate of d , deno ted ˆ d c , can b e summ arized as follows (see [ 10] fo r de ta ils) ˆ d c = ( f − 1 ( S ) , if M = P = Q = 0; f − 1 ( ˆ ρS ) , otherwise (4) 4 Qing Miao, Baoqi Huang ∗ and Bing Jia where ˆ ρ = 2 M 2 M + P + Q . 3.3.2 The Generic Channel Model Case In the gener ic channel mod el [10], the r a ndomn ess on the RSS ca n be character ized by a function g ( d ) , deno ting the probab ility that a dir ectional co m munication link exists f rom transmitter to r eceiv e r with distance d . I n p articular, in the log-no rmal model, we can have g ( d ) = Z ∞ k log d r exp − z 2 2 σ 2 dB √ 2 π σ dB dz (5) where k = 10 α/ lo g 10 ; r deno tes a pseu d o transmission range which depend s on th e antenn a gains, the wa velength of the p ropaga ting signal, th e transmission p ower and th e commun ication th r eshold fo r RSS. Let M , P , and Q contin u ously denote the numb ers of common and non-comm on immediate neig hbors associated with two n odes. we can c o mpute their expectations as fol- lows E ( M + P ) = E ( M + Q ) = λ Z ∞ −∞ Z ∞ −∞ × g ( p ( x − x B ) 2 + ( y − y B ) 2 ) dxdy , (6) E ( M ) = λ Z ∞ −∞ Z ∞ −∞ g ( p ( x − x A ) 2 + ( y − y A ) 2 ) × g ( p ( x − x B ) 2 + ( y − y B ) 2 ) dxdy . (7) Then, by generalizing S and f ( d ) to specify the expec- tations of M , P and Q un d er th e generic chann e l mod el in- stead o f the are as defined under the unit disk m odel, we can have th e fo llowing form u las S = Z ∞ −∞ Z ∞ −∞ g ( p ( x − x B ) 2 + ( y − y B ) 2 ) dxdy , (8) f ( d ) = Z ∞ −∞ Z ∞ −∞ g ( p ( x − x A ) 2 + ( y − y A ) 2 ) × g ( p ( x − x B ) 2 + ( y − y B ) 2 ) dxdy . (9) Moreover , by using (5) and ( 9), we can der ive the fo r- mula for f ( d ) unde r the log- normal model. Similar to 3.3. 1, the distance estimate can b e calculated based on th e in verse of f ( d ) ; that is, ˆ d c =      0 , if M = P = Q = 0; f − 1 ( ˆ ρS ) , if f ( d th ) ≤ ˆ ρ ≤ f (0); d th , if ˆ ρS < f ( d th ) (10) where ˆ ρ = 2 M / (2 M + P + Q ) , and d th denotes the longest distance between two neigh b oring nodes. Howe ver, since the closed- form formulae f or f ( d ) and its in verse are hard o r ev e n im possible to o btain, we thus substitute its inv er se by u sing a n appro ximate p iecewise lin- ear f unction, nam ely th a t a linear regression mo del is estab- lished to pre d ict d for each affine segment. 4 The Proposed Fusi on Method In this section, we in tr oduce the new distan ce estimatio n method based on the aforemen tioned RSS-based m ethod a n d connectivity-based m e th od. T o do so, it is n ecessary to un - derstand the statistical distributions of the RSS-based and connectivity-based distance estimates, re sp ectiv ely . As such, a thoroug h theoretical analysis is carried out to in vestigate both of the distance estimation methods. After that, the MLE method can be instantly applied to estimate the d istance, and par ticularly , the Newton-Raphson method is adopted to solve the cor r espondin g likeliho od function. Besides, th e CRLB associated with the p roposed fusion distanc e esti- mation metho d is fo rmulated to fu rther observe its perfor- mance. 4.1 Statistical An a ly sis o f Distance Estimates In w h at follows, the RSS-based and con nectivity-based dis- tance estimation metho ds shall be analyze d by fo rmulating their statistical distributions, which pa ves the way f or clari- fying th e pr oposed distance estimation m ethod. 4.1.1 The RSS-b ased Case In light o f the RSS-based distance estimation metho d pre- sented in Sub section 3.2, it follows f rom Equatio n ( 1) that d can b e fo rmulated as follows d = 10 P R ( d 0 )( dBm ) − P R ( d )( dBm ) 10 α × 1 0 Z 10 α . (11) By rep lacing the RSS-based distance estimate ˆ d R in (2 ), we can h av e d = ˆ d R × 1 0 Z 10 α . (12) Then, defin e ǫ R to be the multiplicative erro r of the RSS- based distance estimate, namely ˆ d R = d × ǫ R , (13) and ǫ R = 10 − Z 10 α . (14) Since ǫ R is dependen t on th e normal variable Z , th eir probab ility density functio ns, den o ted by p ǫ R ( · ) and p Z ( · ) respectively , satisfy the following equ ation p ǫ R ( x ) = p Z ( z ǫ R ( x )) × | z ′ ǫ R ( x ) | , (15) Estimating Distances via Receiv ed Si gnal Strength and Connecti vi t y in Wire l ess Sensor Networks 5 where z ǫ R ( x ) = − 10 α × log 10 x. (16) Then, we can ob tain the p robab ility density function of ǫ R as f ollows p ǫ R ( x ) = 1 √ 2 π σ R exp  − log 2 10 x 2 σ 2 R  1 x ln 10 , (17) where σ 2 R = σ 2 dB (10 α ) 2 . Since the re lationship of ǫ R and ˆ d R is expressed as (13) and the ab ove method, the pr obability density f unction of the RSS-based distan c e estimation, deno ted by p ˆ d R ( · ) , satisfies the f o llowing e quation p ˆ d R ( x ) = 1 √ 2 π σ R exp − log 2 10 x d 2 σ 2 R ! 1 x ln 10 . (18) 4.1.2 The Connectivity-b ased C ase In Subsection 3. 3, we have introd uced the connectivity-based distance estimation method an d a piecewise linear function to appr o ximate the fu nction f ( d ) . Similar ly , we can ap prox- imate f ( d ) by f ( d ) ≈ k d + b, (19) where k and b are constant. Thus accor d ing to (10), the connectivity-based distance estimate ˆ d c satisfies ˆ ρS = k ˆ d c + b , (20) where ˆ ρ = 2 M / (2 M + P + Q ) is a random variable. Then, de fin e ǫ c to be the erro r of the con nectivity-based distance estimate, na m ely ǫ c = ˆ d c − d, (21) and ǫ c = 1 k ( ˆ ρS − f ( d )) . (22) Then, we are inte r ested in the distribution of the err o r base o n the f ormula of ǫ c . Becau se of the variable M , P and Q are mutually indep endent Poisson random variables with means λf ( d ) , λ ( S − f ( d )) , λ ( S − f ( d )) , respectiv ely , and the additivity property of the ind ependen t Poisson rand om variables. The distribution of 2 M + P + Q can be fo r mulated as 2 M + P + Q ∼ P (2 λS ) . (23) In [24], it ha s proo f ed that the Po isson r andom variable with th e mean λ larger than five can approxim a tely equal to a normal distribution with the me an an d variance are equal to λ . The Poisson ran dom variables 2 M an d 2 M + P + Q with means 2 λf ( d ) and 2 λS satisfy above con d ition, thenc e the Poisson random variables can app r oximately be p resented as f ollows 2 M ∼ N (2 λf ( d ) , 2 λf ( d )) , (24) 2 M + P + Q ∼ N (2 λS, 2 λS ) . (25) Next, in o rder to an alyse th e distribution of the ˆ ρ = 2 M / (2 M + P + Q ) , we consider the r atio of two ind epen- dent normal random variables. In [25], it has proo f ed that the two in depend ent no r mal variables X and Y with mean s and variances ( µ x , σ 2 x ) and ( µ y , σ 2 y ), respectively . The rand om variable Z = X/ Y co u ld be ap proxima ted to the n ormal distribution with th e mean and variance µ z = µ x µ y , σ 2 z =  µ x µ y  2  σ x µ x  2 +  σ y µ y  2 ! . (26) By (24), (25) and ( 26), we can have the statistical distribu- tion o f ˆ ρ satisfies that ˆ ρ ∼ N  f ( d ) S , f 2 ( d ) S 2  1 2 λf ( d ) + 1 2 λS  . (27) According to the fo rmal in (2 2), we can obtain the er- ror d istribution of the con nectivity-based distance estima- tion w h en S and f ( d ) are co nstant ǫ c ∼ N (0 , σ 2 c ) , (28) where σ 2 c = f 2 ( d ) k 2  1 2 λf ( d ) + 1 2 λS  . Therefo re, the probability density functio n o f the err o r ǫ c , denoted by p ǫ c ( · ) , satisfies th e fo llowing equatio n p ǫ c ( x ) = 1 √ 2 π σ c exp  − x 2 2 σ 2 c  , (29) and then, the probab ility density function, denoted by p ˆ d c ( · ) , satisfies the following equ a tio n p ˆ d c ( x ) = 1 √ 2 π σ c exp  − ( x − d ) 2 2 σ 2 c  . (30) 4.2 The Pr o posed Distanc e Estimation Method Giv en the error distributions of the RSS-based and connectivity- based distance estimates, the MLE can b e applied to fuse the distanc e estimate s by th e above two m ethods. Since the distance estimates ˆ d R and ˆ d c rely on different sources of in- formation , we assume th at ˆ d R and ˆ d c are ind ependen t fro m each othe r and express the likelihood function as follows L ( d ) = p ˆ d R ( x 1 ) × p ˆ d c ( x 2 ) = 1 2 π σ R σ c exp − log 2 10 x 1 d 2 σ 2 R − ( x 2 − d ) 2 2 σ 2 c ! 1 x 1 ln 10 . 6 Qing Miao, Baoqi Huang ∗ and Bing Jia (31) Then, th e natu ral log arithm of the likeliho od fun ction is ln L = − ln(2 π σ R σ c x 1 ln 10) − log 2 10 x 1 d 2 σ 2 R − ( x 2 − d ) 2 2 σ 2 c . (32) In order to o btain the maximu m value of ln L , the Newton- Raphson method is ad opted to der i ve the r o ot of the first deriv ative of ln L , deno ted by F ( ˆ d ) = log 10 x 1 ˆ d σ 2 R ln 10 + ˆ d ( x 2 − ˆ d ) σ 2 c . (33) T o do so, let F ′ ( ˆ d ) be the deriv ativ e func tio n of F ( ˆ d ) , and the specific steps of su c c essi vely find in g better approxim a- tions to th e ro ot of the f unction F ( ˆ d ) in clude 1. Select a n initial gu ess d 0 , where ˆ d 0 = ( x 1 + x 2 ) / 2 ; 2. Calculate the values of the function F ( d k ) and d eriv ative F ′ ( d k ) ; 3. Update ˆ d k +1 by using th e iterative e q uation ˆ d k +1 = ˆ d k − F ( ˆ d k ) F ′ ( ˆ d k ) ; 4. Repeat Step 2 an d 3 u n til | ˆ d k +1 − ˆ d k | < ξ , w h ere ξ is a sufficiently small p ositiv e num ber . 4.3 CRLB The CRLB expresses a lower boun d on the v ar iance of any unbiased estimator [26, 2 7] and is equal to the inv erse of the correspond ing Fisher Inform ation Matrix (FIM). In this subsection, the CRLB regarding the p roposed distance esti- mation pro blem, n a m ely estimating the d istance d f rom M , P , Q and the n oisy RSS measureme n t, is formulated u nder the log-no rmal mo del. Specifically , with the un known pa- rameters d and λ , the associated FIM, d enoted FIM ( d, λ ) , is formu late d as FIM ( d, λ ) = λf ′ ( d ) 2 ( 1 f ( d ) + 2 S − f ( d ) ) + κ d 2 − f ′ ( d ) − f ′ ( d ) 2 S − f ( d ) λ ! , (34) where κ =  10 α σ dB ln 10  2 . The detailed d eriv ation o f (34) can be f ound in App endix A. As was proved in [10], f ( d ) is a first order differentiable function , such that the CRLB for d by using any unbiased estimator, deno ted CRLB ( d ) , can be f ormulated as follows CRLB ( d ) =  2 λS 2 ( f ′ ( d )) 2 f ( d )(2 S − f ( d ))( S − f ( d )) + κ d 2  − 1 . (35) W ith the CRLB, the com parison will be made in the ex- perimental analyses section to verif y the effectiveness of the propo sed m ethod. 5 EXPERIMENT AL ANAL YSES In this section, we aim to investigate the accur a cy of the propo sed distance estimation method, and fu rther an alyse the influences of different factors through bo th simulativ e and pr actical measu rements. 5.1 Simulativ e Analyses In the nu merical simu la tio ns, the root-m e a n-square error (RMSE), which equals to the square ro ot o f the squared biases plus variances of the errors in distance estimates, is evaluated in Matlab to measure th e accur acy of the prop osed method. Moreover , the CRLB and the RMSE produc e d by the RSS- based method and the co nnectivity-based metho d a re also calculated in the simulations. The simulative par ameters in relatio n to the WSNs a n d wireless chan nels are describ ed below . 1. The mean value o f RSS m easurements at the reference distance, i.e. P R ( d 0 ) , is − 37 . 47 d Bm; 2. The minimum acceptable RSS value is − 10 0 dBm; 3. Each WSN is de p loyed in a square r egion , the side len gth of which varies with different co nfiguratio ns of α and σ dB ; 4. The sensors are deployed u n der th e ran dom and un iform distribution of mean λ ; 5. The RMSE in each case is evaluated after simulating 10000 d istan ce estimates; 6. µ is the expected number of immed iate n e ig hbors of a sensor , namely µ = E ( M + P ) = E ( M + Q ) , and will take different values given various configu rations o f σ dB and α . For better p resentation, the conn e cti vity ind ex µ will b e used in the following discussion s instead of the sen sor d ensity λ . T o analyz e the error characteristics of th e p r oposed meth od, the influen ces of d ifferent factors, including the expected number of immed iate neighbo rs of a sensor, the variance of shadowing ef fect and the path lo ss expone n t, are inves - tigated in what follows. Firstly , the effect of the expected number of imme d iate neighbo rs o f a senso r (i.e. µ ) on the RMSE is con sidered. Giv en σ dB = 4 an d α = 4 , Fig. 2 depicts the RMSE with µ varying f rom 10 to 40 . As can be seen, we can observe that – given two nearby sensor s, the perform ance of the pro- posed method appro aches to that of the RSS-based dis- tance estimate, and is supe rior to that of the c o nnectivity- based method ; on the co n trary , given two sensors far away from each other , the per formanc e of the proposed method is close to that of the conn ectivity-based metho d and is mu ch better than the RSS-based method ; that is to say , the proposed method alw ay s outper forms the othe r two m e th ods; Estimating Distances via Receiv ed Si gnal Strength and Connecti vi t y in Wire l ess Sensor Networks 7 – the RMSE of the prop osed meth od is slightly higher than the CRLB when two sensors ar e not far way fro m each other, and can even be smaller th an the CRLB due to th e fact that the b ound a ry info r mation (i. e. the up per bound on the distan ce estimate) is intro duced. Secondly , the influ ence o f the noise lev el o n the RMSE is invest igated. As shown in Fig. 3, the RMSE and CRLB are plotted given µ = 20 , α = 4 and σ dB is n umber varying from 4 to 8 , and it ca n be con c luded that – with σ dB increasing, th e perform ance of th e RSS-based methods deter iorates, which is on ac c ount of the incr eas- ing n oises in RSS m e asurements, whereas the pr oposed method and the con nectivity-based m e thod in cur slight changes, w h ich is also con sistent with the CRLB; – the overall p erforma n ce of the pro p osed m ethod is also better than th e oth er two methods. Finally , the ef fect of the path loss exponent is stud ied. As illustrated in Fig. 4, the RMSE and CRLB ar e p lotted giv en µ = 20 , σ dB = 4 and α varying fr om 3 to 6 . It can be seen th a t – with α in creasing, both the RMSE of these three meth- ods and the CRLB decr ease, which is because the com- munication range d ecreases; – the pr oposed method always outperform s th e other two methods and ap proach es to the CRLB. 5.2 Implemen ting the Meth od in Practice T o b etter demon stra te its sup eriority , th e propo sed metho d is implem ented using pr actical RSS m e a surements and de- ployment in formation provided in [28]. Specifically , a WSN consisting of 44 sensors was dep loyed in a real environment as shown in Fig. 5, and the RSS measuremen ts between any two sensors were rep orted. On the basis of their RSS mea- surements, the pro posed metho d can be run to pr oduce prac- tical distance estimates. According to [ 28], define α = 2 . 3 , σ dB = 3 . 92 , P R ( d 0 ) = − 37 . 47 dBm, and the minimum value o f the RSS mea su re- ments is − 55 d Bm. T o a void b ounda ry effects as m uch as possible, we consider th e four sen so rs near the cen ter of the deployment region, i.e. senso r s 15 , 23 , 24 and 25 . The dis- tance estimates amon g these four sensors are calculated by RSS-based, conn ectivity-based and prop osed methods, re- spectiv ely , and are listed in T ab . 1. A s can be seen, th e erro rs of the pro posed metho d are always less than the correspo nd- ing errors ob tained by the other two meth ods. −5 0 5 10 −2 0 2 4 6 8 10 12 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 X Coordinate (m) Y Coordinate (m) Fig. 5 The layout of the sensors in [28] T o sum u p, the pro p osed method is able to ach ieve mo re accurate distance estimates than both the RSS-based method and the connectivity-based method und e r various simu lativ e and practical environments, which confirm s the effecti ve- ness of the prop o sed metho d . 6 CONCLUSIONS In this p aper, we fused the RSS measurements a nd local con- nectivity betwee n two neighb o ring nodes and impleme nted the lo w- cost and accurate distan ce estimatio n metho d. The advantages o f the propo sed method lie in the f ollowing as- pects. Firstly , th e p ractical log-n ormal model was applied to d e d uce the e r ror character istics of the RSS-based m e th od and the co nnectivity-based metho d, which enab les us to fuse two sour ces o f information b a sed th e MLE. Secondly , both simulations an d expe r iments were conducted , and it was shown that th e propo sed m ethod outperf orms its counterp arts and approa c h es to the CRLB in m ost cases. Regarding fu ture works, we would like to apply the pro- posed m ethod with th e existing low-cost localizatio n algo- rithms (e.g. DV -Hop) so as to improve the localization per- forman ce o f WSNs without using extra devices. References 1. Mao, G., Fidan, B., & Anderson, B. D. (2007 ). Wireless sensor net- work localization techniques. Computer networks, 51(10), 2529- 2553. 2. Li, L., Halpern, J. Y ., Bahl, P ., W ang, Y . M., & W attenhofer , R. (2005). A cone-base d distributed topology-control algorithm for wireless mult i-hop networks . IEEE/A CM Trans acti ons on Net- working (TON), 13(1), 147-159. 3. Siripongwutiko rn, P ., & Thipakorn , B. (2008) . Mobility-awar e topology control in mobile ad hoc netwo rks. Computer Commu- nications, 31(14), 3521-3532 . 4. Sitanayah, L., D at t a, A., & Cardell-Oliv er , R. (2010). Heuristic al- gorithm for finding boundar y cyc l es in location-fr ee low density wireless sensor networks . Computer Networks , 54(10), 1630-1645. 8 Qing Miao, Baoqi Huang ∗ and Bing Jia 0 20 40 60 80 0 2 4 6 8 10 12 14 16 18 20 d(m) RMSE(m) Connectivity RSS Proposed CRLB (a) RMSE ( µ = 10 ) 0 20 40 60 80 0 2 4 6 8 10 12 14 16 18 20 d(m) RMSE(m) Connectivity RSS Proposed CRLB (b) RMSE ( µ = 20 ) 0 20 40 60 80 0 2 4 6 8 10 12 14 16 18 20 d(m) RMSE(m) Connectivity RSS Proposed CRLB (c) RMSE ( µ = 30 ) 0 20 40 60 80 0 2 4 6 8 10 12 14 16 18 20 d(m) RMSE(m) Connectivity RSS Proposed CRLB (d) RMSE ( µ = 40 ) Fig. 2 The RMSE and CRLB give n α = 4 , σ dB = 4 and µ = 10 , 20 , 30 , 40 . T ab le 1 The error s by differe nt distance estimation methods Sensor pair (15, 23) (15, 24) (1 5, 25) (23 , 24) (23 , 25) (24, 25) The proposed method 0.603 0.520 0.202 0.776 0.963 0.313 The RSS-based method 0.974 1.168 0.442 1.419 1.654 0.745 The connecti vity-based method 0.615 0.530 0.778 0.828 1.257 0.905 T rue distance 2.286 2.656 1.649 2.781 2.891 1.371 5. Huang, B., W u, W ., & Zhang, T . (2013, October). An improv ed connecti vity-based boundary detection algorithm i n wireless sensor network s. In Local Computer Networks (LCN), 2013 IEEE 38th Confere nce on (pp. 332-335). IEEE. 6. Chitte, S. D., & Dasgupta, S. (2008, October). Distance estimation from receiv ed signal strength under log-normal shado wi ng: Bias and v ariance. In Signal Processing, 2008. ICSP 2008. 9th Interna- tional Conference on (pp. 256-259) . IEEE. 7. Benkic, K., Malajner , M., Planinsic, P ., & Cucej, Z. (2008, June). Using RSSI value for distance estimati on i n wireless senso r net- works based on Zi gBee. In Systems, signals and image process- ing, 2008. IWSSIP 2008. 15th international confere nce on (pp. 303- 306). IEEE. 8. Della Rosa, F ., Pelosi, M., & Nurmi, J. (2012) . Human-induced ef- fects on rss ranging measurements for cooperati ve positioning. In- ternational Journal of Navigation and Observ ation, 2012. 9. Heurtefeu x, K., & V a lois, F . (2012, March). Is RSSI a good choice for localization in wireless sensor network? . In Adv anced Informa- tion Networking and Applications (AINA) , 2012 IEEE 26th Inter- national Conference on (pp. 732-739). IEEE. 10. Huang, B., Y u, C., Anders on, B., & Mao, G. (2014). Estimating distances via connecti vit y in wireless sensor networks . Wireless Communications and Mobile Computing, 14(5), 541-556. 11. Botta, M., & Simek, M. (2013). Adaptive distance estimation based on RSSI in 802.15. 4 network. Radioengineering, 22(4), 1162-11 68. 12. Zanella, A., & Bardella, A. (2014). RSS-ba sed ranging by mul- tichannel RSS av eraging. IEEE Wireless Communications Letters, 3(1), 10-13. 13. Nagpal, R., Shrobe, H., & Bachrach, J. (2003). Organ i zing a global coordina te system from local information on an ad hoc sen- sor network. In Infor mati on processing in sens or networks (pp. 553- 553). Springer Berlin/Heidelber g. 14. Shang, Y ., Ruml, W ., Zhang, Y ., & Fromherz , M. P . (2003, June). Localization from mere connectivity . In Procee dings of the 4th A CM international symposium on Mobil e ad hoc networking & computing (pp. 201-212) . ACM. 15. Buschman n, C., Pfisterer , D., & Fischer , S. (2006, September). Estimating distances using neighbor hood intersection. In Emerging T echno logies and Factory Automation, 2006. ETF A ’06. IEEE Con- ferenc e on (pp. 314-321). IEEE. 16. Buschman n, C., Hellbrck, H., Fische r , S., Kr?ller , A., & Fek ete, S. P . (2007, January) . Radio propagation-a ware distance estimation based on neighborhoo d comparison. In EWSN (pp. 325-340). 17. Li, X. (2007). Collabora t iv e localization with recei ved-signal strength in wireless sensor networks. IEEE T ransac t ions on V ehic- Estimating Distances via Receiv ed Si gnal Strength and Connecti vi t y in Wire l ess Sensor Networks 9 0 20 40 60 80 0 5 10 15 20 25 d(m) RMSE(m) Connectivity RSS Proposed CRLB (a) RMSE ( σ dB = 5 ) 0 20 40 60 80 0 5 10 15 20 25 30 35 d(m) RMSE(m) Connectivity RSS Proposed CRLB (b) RMSE ( σ dB = 6 ) 0 20 40 60 80 0 5 10 15 20 25 30 35 40 d(m) RMSE(m) Connectivity RSS Proposed CRLB (c) RMSE ( σ dB = 7 ) 0 20 40 60 80 0 5 10 15 20 25 30 35 40 45 Connectivity RSS Proposed CRLB (d) RMSE ( σ dB = 8 ) Fig. 3 The RMSE and CRLB give n α = 4 , µ = 20 and σ dB = 5 , 6 , 7 , 8 . ular T echnology , 56(6), 3807-3817. 18. Ouyang, R. W ., W ong, A. K. S., & Lea, C. T . (2010). Receiv ed sig- nal strength-based wireless localization via semidefinite program- ming: Noncooperati ve and cooperativ e schemes. IEEE Tra nsactions on V eh icular T echnology , 59(3), 1307-1318. 19. So, H. C., & Lin, L. (2011). Li near least squares approach for accura te recei ved signal strength based source localization. IEEE T ransac tions on Si gnal Processing, 59(8), 4035-4040. 20. Patwar i, N., Hero, A. O., Perkins, M. , Correal, N. S., & O’ dea, R. J. (2003). Relative location estimation i n wireless sensor networks. IEEE Transa ctions on signal process i ng, 51(8), 2137-2148. 21. Cheng, Y . Y ., & Lin, Y . Y . (2009). A ne w recei ved signal strength based location estimation scheme for wi reless sensor network. IEEE Transa ctions on Consumer Electronics, 55(3). 22. Botta, M., & Simek, M. (2013). Adaptive distance estimation based on RSSI i n 802.15. 4 network . Radioengineering, 22(4), 1162-11 68. 23. Francesc hetti, M. , & Meester , R. (2008) . Random networks for communication: from statistical physics to information systems (V ol. 24). Cambridge Univer sit y Press. 24. Griffi n, T . F . (1992). Distribution of the ratio of two poisson ran- dom variab les (Doctoral dissertation). 25. Dłaz-Francs , E., & Rubio, F . J. (2013). On t he existence of a nor- mal approximation to the distribution of t he ratio of two indepen- dent normal random variables . Statistical Pape rs, 1-15. 26. Salman, N. , M aheshw ari, H. K., Kemp, A. H., & Ghogho, M. (2011, June). Ef fects of anchor placement on mean-CRB for lo- calization. In Ad hoc networking workshop (Med-Hoc-Net), 2011 The 10th IFIP annual mediterranean (pp. 115-118) . IEEE. 27. Ling, Y ., Alexande r, S., & Lau, R. (2012, March). On quantifica- tion of anchor placement. In INFOCOM, 2012 Proceedings IEEE (pp. 2192-2200). IEEE. 28. Patw ari, N. W i reless sensor network localization measurement repository , 2006. A The Derivation of the CRLB The probability d ensity functions of M , P , Q and th e RSS can b e fo rmulated as follows p M ( x 1 ) = ( λf ( d )) x 1 x 1 ! exp( − λf ( d )) , p P ( x 2 ) = ( λ ( S − f ( d ))) x 2 x 2 ! exp( − λ ( S − f ( d ))) , p Q ( x 3 ) = ( λ ( S − f ( d ))) x 3 x 3 ! exp( − λ ( S − f ( d ))) , p P R ( x 4 ) = 1 √ 2 π σ dB exp  − ( x 4 − P R ( d 0 ) + 1 0 α log 10 d ) 2 2 σ 2 dB  , where P R is a rando m variable represen tin g RSS in dBm, and is nor m ally distributed with the mean P R ( d 0 ) − 10 α log 10 d and variance σ 2 dB . 10 Qing Miao, Baoqi Huang ∗ and Bing Jia 0 50 100 150 200 250 0 10 20 30 40 50 60 70 80 d(m) RMSE(m) Connectivity RSS Proposed CRLB (a) RMSE ( α = 3 ) 0 20 40 60 80 0 2 4 6 8 10 12 14 16 18 20 d(m) RMSE(m) Connectivity RSS Proposed CRLB (b) RMSE ( α = 4 ) 0 10 20 30 40 0 1 2 3 4 5 6 7 8 d(m) RMSE(m) Connectivity RSS Proposed CRLB (c) RMSE ( α = 5 ) 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 3.5 4 d(m) RMSE(m) Connectivity RSS Proposed CRLB (d) RMSE ( α = 6 ) Fig. 4 The RMSE and CRLB give n σ dB = 4 , µ = 20 and α = 3 , 4 , 5 , 6 . According to th e p robability de n sity f u nctions, it is straight- forward to form ulate the likelihood functio n as L ( d, λ ) = p M ( x 1 ) × p P ( x 2 ) × p Q ( x 3 ) × p P R ( x 4 ) = λ x 1 + x 2 + x 3 1 √ 2 π σ dB ( f ( d )) x 1 ( S − f ( d )) x 2 + x 3 x 1 ! x 2 ! x 3 ! × e xp  − λ (2 S − f ( d )) − ( x 4 − P R ( d 0 ) + 1 0 log 10 d ) 2 2 σ 2 dB  . And th e n atural lo garithm of the likelihood function can be expressed as follows ln L ( d, λ ) = ( x 1 + x 2 + x 3 ) ln λ + ln  ( f ( d )) x 1 ( S − f ( d )) x 2 + x 3 x 1 ! x 2 ! x 3 ! √ 2 π σ dB  − λ (2 S − f ( d )) − ( x 4 − P R ( d 0 ) + 1 0 α log 10 d ) 2 2 σ 2 dB . Therefo re, the FIM is defin ed as FIM ( d, λ ) = −   E  ∂ 2 ln L ∂ d 2  E  ∂ 2 ln( L ) ∂ λ∂ d  E  ∂ 2 ln( L ) ∂ λ∂ d  E  ∂ 2 ln( L ) ∂ λ 2    . Since f ( d ) ap proxima tes a linear fu nction, the par tial deriv ative can be formu late d as ∂ ln L ∂ d = x 1 f ′ ( d ) f ( d ) − ( x 2 + x 3 ) f ′ ( d ) S − f ( d ) + λf ′ ( d ) − x 4 − P R ( d 0 ) + 1 0 α log 10 d ) σ 2 dB 10 α d ln 10 , ∂ ln L ∂ λ = x 1 + x 2 + x 3 λ − (2 S − f ( d )) , ∂ 2 ln L ∂ d 2 = − x 1 ( f ′ ( d )) 2 f 2 ( d ) − ( x 2 + x 3 )( f ′ ( d )) 2 ( S − f ( d )) 2 − κ d 2 + ( x 4 − P R ( d 0 ) + 10 α log 10 d ) σ 2 dB 10 α d 2 ln 10 , ∂ 2 ln L ∂ λ 2 = − x 1 + x 2 + x 3 λ 2 , ∂ 2 ln L ∂ λ∂ d = f ′ ( d ) . Because M , P , Q and P R are indep endent ran dom vari- ables with m eans λf ( d ) , λ ( S − f ( d )) , λ ( S − f ( d )) and Estimating Distances via Receiv ed Si gnal Strength and Connecti vi t y in Wire l ess Sensor Networks 11 P R ( d 0 ) − 1 0 α log 10 d , respectively , we can have E  ∂ 2 ln L ∂ d 2  = − λf ′ ( d ) 2  1 f ( d ) + 2 S − f ( d )  − κ d 2 , E  ∂ 2 ln L ∂ λ 2  = − 2 S − f ( d ) λ , E  ∂ 2 ln L ∂ λ∂ d  = f ′ ( d ) , where κ =  10 α σ dB ln 10  2 . Then, the FIM can be expressed as follows FIM ( d, λ ) = λf ′ ( d ) 2  1 f ( d ) + 2 S − f ( d )  + κ d 2 − f ′ ( d ) − f ′ ( d ) 2 S − f ( d ) λ ! , and thus, we can have CRLB ( d ) =  2 λS 2 ( f ′ ( d )) 2 f ( d )(2 S − f ( d ))( S − f ( d )) + κ d 2  − 1 .