International   Journal   of   Electrical   and   Computer   Engineering   (IJECE) V ol.   10,   No.   5,   October   2020,   pp.   5535 5545 ISSN:   2088-8708,   DOI:   10.11591/ijece.v10i5.pp5535-5545 r 5535 Err or bounds f or wir eless localization in NLOS en vir onments Omotay o Oshiga, Ali Nyangwarimam Obadiah Department of Electrical and Electronics Engineering, Nile Uni v ersity of Nigeria, Nigeria Article Inf o Article history: Recei v ed Jan 15, 2020 Re vised Apr 18, 2020 Accepted Apr 30, 2020 K eyw ords: Cram ` er -Rao lo wer bound Error analysis Non-parametric Estimation Position error bound W ireless sensor netw orks ABSTRA CT An ef ficient and accurate method to e v aluate the fundamental error bounds for wireless sensor localization is proposed. While there already e xi st ef ficient tools lik e Cram ` er - Rao lo wer bound (CRLB) and position error bound (PEB) to esti mate error limits, in their standard formulation the y all need an accurate kno wledge of the statistic of the ranging error . This requirement, under Non-Line-of-Sight (NLOS) en vironments, is impossible to be met a priori. Therefore, it is sho wn that collecting a small number of samples from each link and applying them to a non- parametric estimator , lik e the g aussian k ernel (GK), could lead to a quite accurate reconstruction of the error dis- trib ution. A proposed Edge w orth Expansion method is emplo yed to reconstruct the error statistic in a much more ef ficient w ay with respect to the GK. It is sho wn that with this method, it is possible to get fundamental error bounds almost as accurate as the theoretical cas e, i.e. when a priori kno wledge of the error distrib ution is a v ailable. Therein, a technique to determine fundamental error limits–CRLB and PEB–onsite without kno wledge of the statistics of the ranging errors is proposed. Copyright c 2020 Insitute of Advanced Engineeering and Science . All rights r eserved. Corresponding A uthor: Omotayo Oshig a Nile Uni v ersity of Nigeria, Ab uja, Nigeria Email: ooshig a@nileuni v ersity .edu.ng 1. INTR ODUCTION Recently , wireless sensor localization ha v e been widely used for positioning and na vig ation with v ar - ious applications in health, transport, en vironment and other commercial services [1, 2, 3, 4]. As we kno w , WSNs comprises numerous of wirelessly connected sensors, as a result sensor positioning has become an im- portant problem. The global positioning system (GPS) currently a v ailable is e xpensi v e, and therein relati v ely fe w sensors are equipped with GPS recei v ers called reference de vices, whereas the other sensors are blind- folded de vices (nodes). Se v eral methods ha v e been proposed to estimate the positions of sensor nodes in WSN, a problem kno wn as Node Localization [5, 6]. Inherently , obtaining the lo wer bound on location errors in relation to e v ery node is an essential and basic problem within the positioning conte xt of WSN. As a result, the most commonly used tool is the Cram ` er - Rao lo wer bound [7, 8, 9, 10], describing the a v erage me an square error (i.e. the distance between the true and estimated node location). Also, it establishes the minimum root mean square error theoretically achie v able with an unbiased estimator and it is commonly used as a designing tool, in the sense that it of fers a bench mark ag ainst which estimat ion algorithms can be compared with. Another popular tool is the position error bound [11, 12, 13] which illus trates the confidence r e gion where a node should be located with a certain confidence interv al. It is important to note that both the CRLB and the PEB are obtained from the fisher information matrix. Since the y both rely on the kno wledge of the distrib ution of the ranging error , which in turn depends on en vironmental and technological f actors, obtaining their formulation a priori is almost impossible, especially in WSNs af fected by mainly Non-Line-of-Sight (NLOS). J ournal homepage: http://ijece .iaescor e .com/inde x.php/IJECE Evaluation Warning : The document was created with Spire.PDF for Python.
5536 r ISSN: 2088-8708 Mainly , there e xists tw o methods for e v aluating the distrib ution of ranging error mea surements- the parametric method which are used for specific and e xplicit distrib utions such as Gaussian, Exponential, Rayleigh etc. and non-parametric method are used for all other distrib utions without e xplicit e xpression. The feasible solution is to approximate the distrib ution statistics of the ranging errors on-site, by collecting ranging samples from each tar get-anchor link and then estimating the lo wer bound on the location errors e v en before tar get localiztion. One immediate application of on-site estimation of error statistics is that this can be used to inform cooperati v e localization algorithms on which nodes to cooperate with to reduce the commulati v e localization error for an y tar get. T o this end, the well kno wn maximum lik elihood par ametric appr oac h is going to f ail, gi v en that in general there is no a priori kno wledge on the error distrib ution. A truly non-parametric approach is therefore re- quired in this case; in particular the k ernel method is v ery appreciated for its capability to reconstruct empirical distrib utions from samples, and in particular i ts Gaussian k ernel (GK) realization. Numerous w orks ha v e been done on error analyses for wireless localization with most ef forts based on Line-of-Sight conditions [14, 15, 16], which lead to se v ere de gradations as NLOS conditions are more appropriate for an accurate wireless localiza- tion. V arious localization algorithms and performance analyses for NLOS en vironment ha v e been proposed [15, 16, 17, 18]. The parametric e xponential distrib ution-based CRLB model in [15] can not be used for other para- metric dist rib utions to simulate NLOS ranging errors. The CRLB in [16] w as deri v ed for NLOS en vironment using on a single reflection model, and can not be used in a situation where most signals arri v e at the recei v er after multi-reflections. The CRLB with or without NLOS statistics w as deri v ed for NLOS situation in [17]. F or the case without NLOS statistics, the authors computed the CRLB in a mix ed NLOS/LOS en vironment and pro v ed that the CRLB for a mi x ed NLOS/ LOS en vironment depends only on LOS signals, while for the case with NLOS statistics, the authors only pro vided a definition of CRLB. In this article, the GK method utilised to obtain the on-s ite the statistic of the ranging errors is reproduced and both the CRLB and PEB are then re written, along with their performance analysis in v ari- ous forms. Compared with the pre vious performance studies for LOS and NLOS conditions, the contrib utions of this article are as follo ws: a. A mathematical description of t he system model and standard error bounds are formulated, which de- picted that the ranging model and bounds deri v ed are applicable to an y distrib ution of ranging errors. F or easy modelling of NLOS conditions, the nakag ami distrib ution model w as used Section 2. b . A Gaussian k e rnel (GK) method w as introduced and a mathem atical formulation of its lo wer bounds were obtained to deri v e the statistical dis trib ution of the errors similar to [18] Section 3. Also, a ne wly proposed Edge w orth e xpansion (EE) method w as introduced and a mathematical formulation of its lo wer bounds were obtained to deri v e the statistical distrib ution of the errors Section 4. c. A thorough and compl ete analyses of CRLBs and PEBs for the GK and EE me thods, which upholdss the proposed EE method by e xhibiting that it indeed comes v ery close in achie ving the fundamental lo wer bound in terms of location error . Its greater ef ficienc y is further pro v ed by the much lo wer number of samples needed to reach the same le v el of accurac y as the GK technique Section 5. 2. SYSTEM MODEL AND FORMULA TION 2.1. System model Consider a netw ork of N nodes in an -dimensional Euclidean space, out of which blindfolded de vices inde x ed 1 ; ; N t ha v e no kno wledge of their location (henceforth tar g ets ), while de vices inde x ed N t + 1 ; ; N t + N a are anc hor s , i.e. reference de vices of a priori kno wn location. F or the sak e of clar - ity , we shall hereafter scrutinize the case of when = 2 , with the remark that the analysis to follo w can be straightforw ardly e xtended to > 2 . The localization problem consists of estimating the location of tar get nodes, gi v en the kno wledge on the location of anchor nodes, and a set of measures of distances amongst de vices typically af fected by errors [8]. T o elaborate, let the position of the i -th de vice be denoted by ( x i ; y i ) , such that the coordinate v ector of the tar get to be approximated is described as , [ x ; y ] = [ x 1 ; ; x N t ; y 1 ; ; y N t ] (1) Int   J   Elec   &   Comp   Eng,   V ol.   10,   No.   5,   O c tober   2020   :   5535   –   5545 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 r 5537 Lik e wise, we describe the anchors’ coordinate v ector by , [ x ; y ] = [ x N t +1 ; ; x N t + N a ; y N t +1 ; ; y N t + N a ] (2) It is well kno wn that when tw o nodes are able to e xchange information, the y are able to estimate the mutual distances between themselv es, a process referred to as r anging . Consistently , ranging measurements are al w ays af fected by noise and often the y are not obtained o v er a LOS link between nodes. In NLOS scenarios, an additional ranging error referred to as bias in the form of a positi v e de viation from the true mutual distance appears. Under these assumptions, the ranging model applicable to a pair of de vices i -th and j -th is gi v en by ~ d ij = d ij + n ij + b ij = q ( x i x j ) 2 + ( y i y j ) 2 + v ij (3) where ~ d ij is the measured distance, d ij is the true distance, n ij is an additi v e white Gaussian noise with mean = 0 and v ariance 2 ij , b ij is the bias, and the r esidual noise v ij where the noise and bias are modelled jointly . 2.2. Standard err or bound f ormulations Here, the fisher information matrix (FIM) J [9] as the fundamental matrix to obtain both the CRLB and PEB are clearly formulated, with the aim of clearly introducing the notations and methods to be emplo yed in the Sections 3. and 4. where the g aussian k ernel (GK) [18] and edge w orth e xpansion (EE) (proposed) [19] error bounds will be formulated and discussed. Let ~ d be the range measurements (measured distances) v ector denoted as ~ d , n ~ d ij o (4) where i; j = 1 : : : N for i 6 = j . Let ^ be an estimate of the v ector parameter and E [ ^ ] as the e xpected v alue of ^ . The CRLB matrix relates to the Fisher information matrix J [9] as E h ( ^ )( ^ ) T i J 1 (5) The Fisher information matrix J is accordingly gi v en as J , E 2 4 @ ln f ( ~ d j ) @   @ ln f ( ~ d j ) @ ! T 3 5 (6) The log of the joint conditional probability density function (PDF) is ln f ( ~ d j ) = N X i =1 X j 2 H ( i ) j <i l ij (7) where l ij = ln f ~ d ij j ( x i ; y i ; x j ; y j ) . Substituting l ij in (7) and in (6), the FIM is then denoted by [14] J , " J xx J xy J xy J y y # (8) where [ J xx ] k l = 8 > > > < > > > : X j 2 H ( k ) E " @ l k j @ x k 2 # e k l E @ l k l @ x k @ l k l @ x l ; [ J xy ] k l = 8 > > > < > > > : X j 2 H ( k ) E @ l k j @ x k @ l k j @ y k e k l E @ l k l @ x k @ l k l @ y l ; Err or bounds for wir eless localization in NLOS en vir onments (Omotayo Oshiga) Evaluation Warning : The document was created with Spire.PDF for Python.
5538 r ISSN: 2088-8708 [ J y y ] k l = 8 > > > < > > > : X j 2 H ( k ) E " @ l k j @ y k 2 # k = l e k l E @ l k l @ y k @ l k l @ y l k 6 = l and k ; l = 1 : : : n are the blindfolded (tar get) nodes. J xx ; J y y ; J xy , and J are of sizes n n and 2 n 2 n , respecti v ely . 2.3. Modeling range measur ements The statistics of the measured distances between nodes- adopting the most recognised propag ation models in mobile and wireless communication in the literature [21, 22], has been modeled after the nakag ami distrib ution (ND). The nakag ami distrib ution w as selected to fit empirical data and is kno wn to pro vide a closer match to most measurement data than either the Gaussian, Rayleigh or Rician distrib utions. Be yond its empirical j u s tification, the nakag ami distrib ution is often used for the follo wing reas on s . First, the nakag ami distrib ution can model en vironmental conditions that are either more or less se v ere than Rayleigh f ading. When the nakag ami shape f actor is 1, the nakag ami distrib ution becomes the Rayleigh distrib ution, and when the nakag ami shape f actor is 1/2, it becomes a one-sided Gaussian distrib ution. Second, the Rice distrib ution can be closely approximated using the close form relationship between the Rice f actor and the nakag ami shape f actor . Due to the empirical data and w ork done in [21], the nakag ami distrib ution w as chosen to model the NLOS conditions for ranging measurements. The PDF of the residual noise v ij , to e v aluate the performance of both the g aussian k ernel and edge- w orth e xpansion methods, will therein be f v ij ( v ij ) = 2 m m ij ij ( m ij ) m ij ij v 2 m ij 1 ij exp m ij ij v 2 ij (9) where m ij and ij are the shape and controlling spread parameters of the Nakag ami distrib ution. 2.4. Bounds deri v ation using nakagami distrib utions Gi v en the obtained ranging model’ s PDF , it is no w attainable to deri v e a ne w formula for the FIM. From (9), tak e its natural log arithm and substitute the result into @ l k l @ x k , @ l k l @ y k , @ l k l @ x l and @ l k l @ y l yields @ l k l @ x k = x k x l d k l 2 m k l v k l k l 2 m k l 1 v k l ; @ l k l @ y k = y k y l d k l 2 m k l v k l k l 2 m k l 1 v k l ; @ l k l @ x l = x k x l d k l 2 m k l v k l k l 2 m k l 1 v k l ; @ l k l @ y l = y k y l d k l 2 m k l v k l k l 2 m k l 1 v k l (10) and therefore [ J xx ] k l = 8 > > > < > > > : X j 2 H ( k ) A k j ( x k x j ) 2 d 2 k j e k l A k l ( x k x l ) 2 d 2 k l l ; [ J xy ] k l = 8 > > > < > > > : X j 2 H ( k ) A k j ( x k x j )( y k y j ) d 2 k j e k l A k l ( x k x j )( y k y j ) d 2 k l ; [ J y y ] k l = 8 > > > < > > > : X j 2 H ( k ) A k j ( y k y j ) 2 d 2 k j e k l A k l ( y k y l ) 2 d 2 k l where A k l = E " 2 m k l v k l k l 2 m k l 1 v k l 2 # = Z 1 1 2 m k l v k l k l 2 m k l 1 v k l 2 f v k l ( v k l ) d v k l (11) Int   J   Elec   &   Comp   Eng,   V ol.   10,   No.   5,   O c tober   2020   :   5535   –   5545 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 r 5539 3. ERR OR ESTIMA TION VIA GA USSIAN KERNEL In [18], using the g aussian k ernel (GK) method the error bound formulation w as obtained to model the PDF of the positi v e de viation - bias b ij . This step w as tak en to ensure that the white Gaussian noise n ij and positi v e bias b ij in the ranging errors were to be modeled independently . In this article, the w ork in [18] w as well modified and impro v ed upon where the residual noise w as modeled independently so as to enable both the noise and bias to be modeled jointly as the residual noise v ij , as it is well kno wn in the literature the impossibity of seperating LOS noise from NLOS bias in a wireless en vironment. 3.1. Err or distrib ution r econstruction The PDF of the residual noise v ij is obtained from samples of ranging measurements . This is an estimation of the true distrib ution by b uilding a sum of k ernels (which are deri v ed from an e xponentially decaying function) of the collected ranging samples, whose ef ficienc y and accurac y depends on the total number of collected samples P . Between the i -th and j -th nodes, S v ij q is defined as the q -th sample o v er the link, the non-parametric Gaussian K ernel technique estimates the PDF of the residual noise as f v ij ( v ij ) = 1 p 2 P h ij P X q =1 exp   ( v ij S v ij q ) 2 2 h 2 ij ! (12) where exp( ) is the Gaussian k ernel e xponential function and the smoothing constant h ij is the width of this Gaussian k ernel function gi v en as 1 : 06 s P 1 = 5 ( s is the sample standard de viation of the residual noise). 3.2. Bounds deri v ation using gaussian k er nel F ollo wing the same approach as in Subsection 2.4., from (12), the natural log arithm can be substituted into @ l k l @ x k , @ l k l @ y k , @ l k l @ x l and @ l k l @ y l obtaining @ l k l @ x k = x k x l d k l g k l ( v k l ) f v k l ( v k l ) ; @ l k l @ y k = y k y l d k l g k l ( v ij ) f v k l ( v k l ) ; @ l k l @ x l = x k x l d k l g k l ( v ij ) f v k l ( v k l ) ; @ l k l @ y l = y k y l d k l g k l ( v k l ) f v k l ( v k l ) (13) where g k l ( v k l ) = 1 p 2 P h ij P X t =1 exp ( v k l S b k l t ) 2 2 h 2 k l v k l S b k l t h 2 k l (14) and the elements of the Fisher Information are similar with (11) e xcept for the coef ficient: A k l = E " g k l ( v k l ) f v ij ( v k l ) 2 # = Z 1 1 g k l ( v k l ) 2 f v k l ( v k l ) d v k l = 1 2 k l k l 2 LOS Z 1 1 g k l ( v k l ) 2 f v k l ( v k l ) d v k l : k ; l 2 NLOS (15) where k l 2 LOS and k l 2 NLOS represent propag ation conditions between nodes k and l . 4. ERR OR ESTIMA TION VIA EDGEW OR TH EXP ANSION The ranging error approximation technique presented in the pre vious section, though rob ust, is con- strained by the enormous amount of samples required to obtain a f air accurac y of the approximates of the distrib ution of a gi v en set of samples. In the follo wing, we introduce a more ef ficient and general method, based on Edge w orth e xpansion, with tw o main adv antages: a much smaller number of samples are required for approximation and the possibility to model both the additi v e Gaussian noise and the positi v e bias jointly . While the prospect of reducing the number of samples required to obtain a f air accurac y can not be o v erem- phasized, it is essential to state that, in wireless channels, the positi v e bias and Gaussian ranging errors cannot Err or bounds for wir eless localization in NLOS en vir onments (Omotayo Oshiga) Evaluation Warning : The document was created with Spire.PDF for Python.
5540 r ISSN: 2088-8708 be separated from each other . Therein, we describe the process of reconstructing the ranging error distrib ution from samples, and then the con v er gence and monotonicty of moments from samples is sho wn, thereby pro v- ing a clear impro v ement in accurac y with respect to the Gaussian k ernel technique, and finally the proposed formulation of PEB and CRLB are sho wn. 4.1. Err or distrib ution r econstruction The Edge w orth e xpansion whi ch is an impro v ed v ersion on the central limit theorem (CL T) is a true asymptotic e xpansion of the PDF of a g aussian v ariable ^ x = ( x ) = in the po wers of t he mean . EE is a formal series of functions that has the characteristics of truncating a series after a finite number of terms, which is suf ficient enough to pro vide an accurate estimation to this function, therein the estimation error is monitored [19]. The EE as a non-parametric approximator can be used for estimating the PDF of gi v en ranging errors from their sample moments w [19]. The EE is gi v en as f ( x ) = N ( ; 2 ) 2 4 1 + 1 X s =1 s X f k w g H e s +2 r ( ^ x ) s Y w =1 1 k w ! S w +2 ( w + 2)! k w 3 5 (16) where, N ( ; 2 ) is the PDF of a normal distrib ution with mean and v ariance 2 , S w +2 = w +2 w +1 2 , w are the cumulants obtained from the sample moments w as s = s ! X f k w g ( 1) ( r 1) ( r 1)! s Y w =1 1 k w ! w w ! k w (17) The set f k w g consists of all non-ne g ati v e (positi v e and zero) inte ger solutions of the Diophantine set of equations s = k 1 + 2 k 2 + + sk s and r = k 1 + k 2 + + k s . The Chebyshe v-Hermite polynomial H e n ( ^ x ) is H e s ( ^ x ) = s ! s= 2 X k =0 ( 1) k ^ x s 2 k k !( s 2 k )!2 k (18) and the mean and v ariance of the ranging errors are = 1 and 2 = 2 , respecti v ely [20, 21, 22]. The sample m oments from the ranging errors are w = 1 =n n X i =1 X i w , where X i are the r anging errors and w = 1 ; 2 ; 3 : : : are the orders of the moment. T o determine the number of orders of moment w required to estimate a gi v en sample, the standard error s of the samples is calculated using 2 s = p P , where s must be 0 : 3 , for each order w . The Edge w orth Expansion is used to model the residual noise v ij , hence, the estimated PDF of the residual noise f v ij ( v ij ) is f v ij ( v ij ) = 1 q 2 2 ij exp   ( v ij ) 2 2 2 ij ! 0 @ 1 + 1 X s =1 s X f k w g A s H e s +2 r ( ^ x ) 1 A (19) where A s = s Y w =1 1 k w ! S w +2 ( w + 2)! k w and v ij = ~ d ij d ij . 4.2. Efficiency and con v er gence of sample moments of the EE method T o illustrate the ef fecti v eness and ef ficienc y of the Edge w orth Method, it is mandartory to dem osntrate the con v er gence of its sample moments as the number of samples P increases [23, 24], and therein compare it with the Gaussian K ernel. Using the Nakag ami Distrib uted random v ariables as seen in Figure 1, the true moments w of a ND ( m = 1 ; = 1 ) is compared with sample moments w [23] for dif ferent number of samples and moment orders w = 1 ; : : : ; 4 . The de viation ^ e w of the sample moments from the true moment Int   J   Elec   &   Comp   Eng,   V ol.   10,   No.   5,   O c tober   2020   :   5535   –   5545 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 r 5541 is obtained as t he absolute of ( w w ) = w . Also, the monotonicity of the sample moments is seen in Figure 2. The K ulback Leiber Di v er gence of the PDFs obtained using the tw o methods (GK and EE) from the true/theoretical Nakag ami PDF is sho wn in Figure 3. F or KLD=0.01, the proposed EE required less than 300 samples, while the GK needs approximately 500 samples to obtain similar results. This delta increases e v en more for lo wer le v el of di v er genc y: to reach a KLD=0.0075, the GK method needs approximately twice the number of samples than the EE method. As a result, the Edge w orth method is a good choice for approximation the distrib ution of ranging errors from samples. 100 200 300 400 500 600 700 800 900 1000 0 0.05 0.1 0.15 0.2 0.25 Nak agam i m = 1 , Ω = 1 Er r or e w Nu m b er of S am p l es P w = { 1 , 2 , 3 , 4 } Figure 1. Con v er gence of sample moments w 1 2 3 4 5 6 7 0 20 40 60   Nak agam i m = 1 , Ω = 1 Mom en t : α w O r d er : k T r u e Mom en t s P = 200 P = 500 P = 1000 Figure 2. Monotonicity of sample moments w 100 200 300 400 500 600 700 800 900 1000 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045   Nak agam i m = 1 , Ω = 1 Ku l l b ac k- L ei b l er D i v er gen ce Nu m b er of S am p l es P PD F - G K PD F - EE Figure 3. CKL Di v er gence of the tw o estimators 4.3. Bounds deri v ation using edgew orth expansion From the formulation of the approximated PDF of the residual error in (19), the creation of the Fi sher information is the same with the GK: This yields @ l k l @ x k = x k x l d k l v k l 2 k l g k l ( v k l ) k l f v k l ( v k l ) ; @ l k l @ y k = y k y l d k l v k l 2 k l g k l ( v k l ) k l f v k l ( v k l ) ; @ l k l @ x l = x k x l d k l v k l 2 k l g k l ( v k l ) k l f v k l ( v k l ) ; @ l k l @ y l = y k y l d k l v k l 2 k l g k l ( v k l ) k l f v k l ( v k l ) (20) where g k l ( v k l ) = 1 q 2 2 ij exp   ( v ij ) 2 2 2 ij ! 1 X s =1 s X f k m g ( s + 2 r ) A s H e s +2 r 1 ( ^ x ) (21) Err or bounds for wir eless localization in NLOS en vir onments (Omotayo Oshiga) Evaluation Warning : The document was created with Spire.PDF for Python.
5542 r ISSN: 2088-8708 The elements of the Fisher Information are similar with the GK e xcept for the coef ficient: A k l = E " v k l 2 k l g k l ( v k l ) k l f v k l ( v k l ) 2 # = Z 1 1 v k l 2 k l g k l ( v k l ) k l f v k l ( v k l ) 2 f v k l ( v k l ) d v k l (22) 5. PERFORMANCE EV ALU A TION Mo ving forw ard from the v arious theoretical analyses presented in this paper , we can state that the EE methods can approximate the statistics of ranging errors (using Nakag ami Distrib ution), with lesser samples and more accurac y with respect to the GK as such we no w consider real netw ork topologies to further illustrate the performances of both methods. Therefore, a re gion of 10 m 10 m is emplo yed, where three ( a n = 3 ) anchors are placed to form a triangular shape and three ( n = 3 ) blindfolded de vices (tar gets), not connected together , are randomly placed within the con v e x of the anchors. The tw o error bounds - the CRLB and the PEB - will be ultilized to e v aluate the performance of the tw o estimators - EE and GK. The a v erage CRLB for an y netw ork topology can be computed using " = 1 n f J 1 g (23) while the PEB can be illustrated by the 95% Confidence Interv al C i = 0 : 95 , whose mathematical formulation is sho wn. The Fisher Ellipse parameters of the i -th tar get i are estimated from the co v ariance matrix i , which is a combination of the error v ariance 2 i : x and 2 i : y on the “x” and “y” dimensions, respecti v ely and t he cross-term i : xy , gi v en as i , " 2 i : x i : xy i : xy 2 i : y # (24) The directions of the scattering in the space for the v ector i are kno wn to be directly proportional to the eigen v alues associated to i up to a f actor of i [25, 11, 12]. In particular , the axis direction of the ellipse which describes this scattering in the space is 2 p i i :1 , 2 p i i :2 , where i :1 , 1 2 h 2 i : x + 2 i : y + q ( 2 i : x 2 i : y ) 2 + 4 2 i : xy i ; i :2 , 1 2 h 2 i : x + 2 i : y q ( 2 i : x 2 i : y ) 2 + 4 2 i : xy i (25) If i : y > i : x , then in (25) the orders of i :1 and i :2 are sw apped. The proportionality f actor i can be related to the confidence interv al C i in that the tar get i is enclosed in an ellipse, as such i = 2 ln(1 C i ) It follo ws that the Fisher Ellipse for the i -th tar get i is described through the follo wing in [25] [( x p i : x ) cos i + ( y p i : y ) sin i ] 2 i i :1 + [( x p i : x ) sin i ( y p i : y ) cos i ] 2 i i :2 = 1 (26) where the r otation angle i describes the of fset between the principal axis for the ellipse and reference axis and it is defined as i , 1 2 arctan   2 i : xy 2 i : x 2 i : y ! – Note that for 2 i : x = 2 i : y , then i = 0 . Figure 4 depicts the le v el of accurac y and ef ficienc y of the approximated CRLBs " , as a function of the sample number P using Nakag ami distrib uted random v ariables with b ij uniformly selected between 0 4 for NLOS and = 0 : 5 for both LOS and NLOS ranging errors within each tar get-to-anchor links. Int   J   Elec   &   Comp   Eng,   V ol.   10,   No.   5,   O c tober   2020   :   5535   –   5545 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 r 5543 0 100 200 300 400 500 600 700 800 900 1000 0.08 0.1 0.12 0.14   Av er age CRL B ¯ ε ( i n m et r es) Nu m b er of S am p l es P CRL B - G K CRL B - EE Th eor et i cal CRL B Figure 4. A v erage CRLB as a function of samples T o compare both estimators, a line representing the theoreti cal CRLB, i.e. computed with per fect kno wledge of the statistic of the propag ation channel has been added to the plots. Clearly the CRLB of recon- structed EE is much closer to the theoretical one than the GK: the EE performs better for an y samples. From the abo v e deri v ed CRLBs, the minimum number of samples P required for obtaining accurate results are analyzed. As sho wn in Figure 3, it is seen that the non-parametric estimators con v er ges to the true PDF with suf ficient sample size, therefore the estimated CRLBs con v er ges quickly to a stable v alue as P increases. Furthermore, the tw o estimators are no w represented by their r especti v e Fisher ellipses (theoretica l and reconstructed from the tw o methods) for P = 50 samples as seen in Figure 5. Clearly , the samples reconstructed with the EE estimator almost perfectly match t he theoretical one, where the y v ary only in the axis orient ation. As the sample number increases to P = 250 samples, the Fisher ellipses of the tw o estimators ha v e almost or matching axis orientation to the theoretical PEB with the EE method much closer than the GK method. T o clearly and better capture the dif ferences between the t w o estimators with respect to the theoretical PEB, Figure 6 depicts, as a function of the number of samples, the inner product PEB , 1 N t N t X i =1 h ^ A i A i i p ^ A i A i ; where hi denotes the inner product, A is the area of the theoretical Fisher ellipse and ^ A is the area of the reconstructed methods (EE or GK method). From the discussion in this section, it can be clearly seen that Edge w orth Expansion method performs f ar better than the Gaussian K ernel method, which therein implies that theapproximated Fisher Ellipses of Edge w orth Expansion are much closer in size and orientation to the theoretical Fisher Ellipses. 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10   Nak agam i b = [ 0 − 4] , σ = 0 . 5 , P = 50 y - co or d i n at es ( i n m et r es) x - co or d i n at es ( i n m et r es) An c h or s T ar get s Th eor et i cal PEB PEB wi t h EE PEB wi t h G K 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10   Nak agam i b = [ 0 − 4] , σ = 0 . 5 , P = 250 y - co or d i n at es ( i n m et r es) x - co or d i n at es ( i n m et r es) An c h or s T ar get s Th eor et i cal CRL B PEB - EE PEB - G K Figure 5. The 95% Fisher ellipses, theoretical, and estimated with P = 50 & 250 samples collected per link Err or bounds for wir eless localization in NLOS en vir onments (Omotayo Oshiga) Evaluation Warning : The document was created with Spire.PDF for Python.
5544 r ISSN: 2088-8708 50 100 150 200 250 300 350 400 450 500 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86   Nak agam i b = [ 0 − 4] , σ = 0 . 5 ∆ M Nu m b er of S am p l es P CRL B - G K CRL B - EE Figure 6. as a function of the number of samples 6. CONCLUSIONS This article clealry focuses on the error analyses of approximating and reconstructing the statistics of the ranging measurements without a priori kno wledge of t he wireless channel. A popular and non-paremetric estimator , the Gaussian k ernel w as first decribed and utilized for the approximation and reconstruction of the error distrib utions from samples and the corresponding error bound w as deri v ed. Futhermore, an Edge w orth Expansion method w as therein, to reconstruct the error distrib ution statistics from samples of the ranging mea- surements by e xploiting the ef ficienc y and ef fecti v eness of moment con v er gence. This approach w as clearly sho wn and pro v en to be v alid for Non-Line-of-Sight conditions, where it is impossible to estimate the statistics of the ranging errors a prior . Res u l ts and figures illustrated sho wed that the Edge w orth e xpansion technique is a f ar more ef ficient and accurate technique than the Gauss ian k ernel method, requiring lesser sample size to reach the samilar le v el accurac y . REFERENCES [1] D. Macagnano, G. Destino, and G. Abreu, ”A comprehensi v e t utorial on localization: algorithms and performance analysis tools, ” International J ournal of W ir eless Information Networks , v ol. 19, no. 4, pp. 290-314, 2012. [2] G. Y an ying, A. Lo , and I. Nieme geers, ”A surv e y of indoor positioning systems for wireless personal netw orks, ” IEEE Communications Surve ys T utorials , v ol. 11, no. 1, pp. 13-32, 2009. [3] L. Hui, H. Darabi, P . Banerjee, and L. Jing, ”Surv e y of wireless indoor positioning techniques and sys- tems, ” IEEE T r ans. on Systems, Man, and Cybernetics, P art C: Applications and Re vie ws , v ol. 37, no. 6, pp. 1067-1080, 2007. [4] N. P atw ari, J. Ash, S. K yperountas, et al, ”Locating the nodes: Cooperati v e localization in wireless sensor netw orks, ” IEEE Signal Pr ocessing Ma gazine , v ol. 22, no. 4, pp. 54-69, 2005. [5] K.-T . Feng, C.-L. C hen, and C.-H. Chen, ”Gale: An enhanced geometry-assisted location estimation algorithm for nlos en vironments, ” IEEE T r ans. on Mobile Computing , v ol. 7, no. 2, pp. 199-213, 2008. [6] Y . Qi and H. K obayashi, ”On geolocation accurac y with prior information in non-line-of-sight en viron- ment, ” IEEE Pr oceedings on the 56th V ehicular T ec hnolo gy Confer ence-F all , v ol. 1, pp. 285-288, 2002. [7] H. V . Poor , ”An introduction to signal detection and estimation 2nd Ed, ” Springer -V erlag Ne w Y ork, 1994. [8] H. V an T rees, ”Detection, Estimation, and Modulation Theory , ” W ile y , 2004. [9] S. Kay , ”Fundamentals Statistical Processg, ” Prentice Hall, 2001. [10] T . W ang, ”Cramer -rao bound for localization with a priori kno wledge on biased range measurements, ” Aer ospace and Electr onic Systems, IEEE T r ansactions on , v ol. 48, no. 1, pp. 468-476, 2012. [11] S. Se v eri , G. Abreu, G. Destino, and D. Dardari, ”Ef ficient and Accurate Local ization in Multihop Net- w orks, ” Pr oc. Asilomar Conf . Signals, Systems, and Computer s, 2009. [12] J. Saloranta, S. Se v eri, D. Macagnano, and G. Abreu, ”Algebraic confidence for sensor localization, ” Pr oc. Asilomar Conf . Signals, Systems, and Computer s, 2012. Int   J   Elec   &   Comp   Eng,   V ol.   10,   No.   5,   O c tober   2020   :   5535   –   5545 Evaluation Warning : The document was created with Spire.PDF for Python.