CramerRao lower bounds for nonhybrid and

EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONS Eur. Trans. Telecomms. (2011) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ett.1530

RESEARCH ARTICLE

Cramer–Rao lower bounds for nonhybrid and hybrid localisation techniques in wireless networks Mohamed Laaraiedh*, Stéphane Avrillon and Bernard Uguen IETR Labs, University of Rennes 1, Campus Beaulieu, Rennes 35042, France

ABSTRACT Theoretical performances of localisation techniques are revisited in this paper. Expressions of asymptotic Cramer–Rao lower bounds are derived for nonhybrid and hybrid localisation techniques based on received signal strength indicator, time of arrival and time difference of arrival under Gaussian uncorrelated measurements assumption. These localisation techniques and the useful expressions of their maximum likelihood estimators are presented in this paper. From these expressions, asymptotic Cramer–Rao lower bounds expressions are derived. Simulations are then carried out using realistic location-dependent parameters (LDPs) values to evaluate different nonhybrid and hybrid techniques of localisation. Simulations show that the fusion of more than one LDP enhances localisation accuracy and gives alternatives for the lack of LDPs. Moreover, the number, the nature, the precision of measurements and the reference nodes positions are the key parameters that influence localisation accuracy. Copyright © 2011 John Wiley & Sons, Ltd. *Correspondence Mohamed Laaraiedh, IETR Labs, University of Rennes 1, Bat 11D, Campus Beaulieu, Rennes 35042, France. E-mail: [email protected] Received 1 June 2010; Revised 28 September 2011; Accepted 28 September 2011

1. INTRODUCTION Localisation systems are more and more requested and implemented in different areas. Proposed location-based services (LBS) are various [1]. These services can be traditional such as localisation and tracking or more sophisticated such as advertising, emergency and security. The accuracy offered by a localisation system must match the quality of service requested by the user and/or the network. Thus, the localisation system must be robust and elaborated enough to offer the requested quality of service. As an example, emergency and security services require usually a high precision of localisation that is not usually the matter for leisure or advertising services. A localisation system must be able to offer each service with the requested accuracy. Nevertheless, the nature of localisation problems affects the localisation accuracy and may prevent the achievement of the requested accuracy. In fact, localisation is usually based on location-dependent parameters (LDPs) measured by the mobile station (MS) and/or the network. These LDPs are mainly received signal strength indicator (RSSI), time of arrival (TOA) and time difference of arrival (TDOA). These LDPs are usually affected by propagation phenomena and channel variability. In addition to this uncertainty coming from LDPs Copyright © 2011 John Wiley & Sons, Ltd.

measurements, the number and the type of used LDPs affect the localisation accuracy. The localisation estimator used to solve the problem and the geometric configuration of radio nodes themselves may affect the localisation accuracy as well [2]. All these factors must be taken into consideration when developing a localisation system and/or service. Localisation techniques can be classified in two groups: nonhybrid techniques and hybrid techniques. In the case of nonhybrid techniques, only one type of LDP is used (RSSI, TOA or TDOA). In the case of hybrid techniques, more than one type of LDP is used [3]. The main advantage of hybrid techniques is overcoming the lack of LDPs. In fact, to be localised in 2D (respectively in 3D) landmark, an MS has to collect at least three (respectively four) measurements. In nonhybrid systems, this amount may not be guaranteed especially for time-based LDPs (TOA and TDOA) because they request ranging techniques that are not implemented in all radio access technologies (RATs). Hence, using different LDPs in nature gives an alternative for the lack problem. This is the concept of hybrid techniques. Moreover, fusion of hybrid LDPs may compensate the imprecision of some measurements by another more precise measurements and thus enhance localisation accuracy offered by the system. Nevertheless, hybrid

Cramer Rao Bounds in Wireless Localization

techniques must be smart enough to fuse different LDPs and enhance localisation accuracy [4]. Evaluating the reachable accuracy of localisation techniques is an important step before defining and implementing localisation systems. The Cramer–Rao lower bound (CRLB) is used to be an accuracy criterion for various estimation problems among the localisation problems [5–9]. In this paper, the asymptotic CRLB (ACRLB) will be used to evaluate nonhybrid and hybrid localisation techniques based on RSSI, TOA and TDOA. For this, we present the development of ACRLBs for these different localisation techniques. Then we study, using these ACRLBs, the effect of different radio and geometric parameters on the achieved localisation accuracy. The rest of the paper is organised as follows. We start, in Section 2, by reviewing nonhybrid and hybrid localisation techniques based on RSSI, TOA and TDOA. Then in Section 3, we derive the ACRLBs for the different techniques presented in Section 2. In Section 4, we present carried out simulations that support our theoretical results and draw conclusions about the performances of different localisation techniques. Finally, concluding remarks and perspectives of our work are presented in Section 5.

2. LOCALISATION TECHNIQUES In this section, we present the main techniques of localisation used in wireless localisation systems. These techniques are mainly based on RSSI, TOA and TDOA. We present also hybrid techniques that consist in fusing two or three from these LDPs. For each LDP, we present the techniques of estimation and measurement, the related modelling and the maximum likelihood (ML) estimator expression. These expressions will be then useful for deriving the expressions of CRLBs. 2.1. Radio signal strength indicator Radio signal strength indicator localisation methods were first introduced in 1969 [10]. RSSI is used to be the easiest and cheapest modality for wireless localisation because it can be obtained at no additional cost with each radio message sent or received [11]. RSSI can be the power (i.e. the magnitude of electrical field) being received by an antenna or the attenuation affecting the signal. The power attenuation is mainly the result of propagation phenomena, fading and channel fluctuations. These propagation phenomena make difficult the prediction of the RSSI in radio propagation channels. Hence, different path loss models are proposed. These path loss models are essential for localisation because they relate RSSI to the distance (i.e. to the location of the device). With the utilisation of these path loss models, RSSI can be easily estimated in each point of space. Nevertheless, the accuracy of RSSI estimation depends strongly on the accuracy of the used path loss model. In this paper, we are particularly interested in the log-normal shadowing model [12]. This model is

M. Laaraiedh, S. Avrillon and B. Uguen

widely used in radio and localisation applications because of its linearity and simplicity. According to this model, the received power Pr , expressed in dB, is given by d Pr D P0 10np log10 C Xsh (1) d0 where P0 is the path loss at the reference distance d0 , which is usually taken equal to 1 m [13]. np is the propagation constant. Xsh , which is modelled as a zero-mean Gaussian random variable with a standard deviation sh , represents the shadowing caused by obstacles between the transmitter (Tx) and the receiver (Rx) that attenuates signal power through absorption, reflection, scattering and diffraction. Giving this modelling of Xsh , the distance dk D kX Xk k between the targeted MS, whose position is X D Œx; yT , and the kth anchor node Xk D Œxk ; yk T respect a log-normal distribution [14]. The probability density function (pdf) of dk is given by

1

pk .dk / D p e 2dk Sk

.ln dk Mk /2 2S 2 k

(2)

where Sk and Mk are path loss parameters of the kth radio link. These parameters are defined respectively by ln.10/sh 10np

(3)

.Pr P0 / ln.10/ C ln.d0 / 10np

(4)

Sk D

Mk D

Assuming independence between measurements, the joint pdf of the K estimated distances is given by p1;:::;K .d1 ; : : : ; dK / D

K Y kD1

1

e p 2dk Sk

.ln dk Mk /2 2S 2 k

(5) and the log-likelihood function is then calculated as the logarithm of p1;:::;K .d1 ; : : : ; dK / and it is given by FRSSI D

K X

2 ln dk Mk Sk2 2Sk2

kD1

ln

p

S2 2Sk Mk C k 2

(6)

O in the ML sense is the Hence, the best position X position X that minimises FRSSI . It is given by O D arg min FRSSI X

(7)

X

More explicitly, this solution satisfies the following condition: @FRSSI D0 (8) rFRSSI D @X O XDX Eur. Trans. Telecomms. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/ett



with

rFRSSI D

K X 1 kD1

Mk Sk2 ln dk

Sk2

dk 2

O Xk / .X

(9)

In this paper, we assume a Gaussian error on TOA (i.e. on ranges). The range rk is hence centred on the true distance dk D kXXk k that separates the kth anchor node and the true position of the terminal. rk N dk ; k2

(10)

2.2. Time of arrival The TOA is defined as the travel time of the signal from the Tx to the Rx. The distance that separates Tx and Rx can be simply obtained by multiplying the TOA by the speed of light, c. Different techniques of ranging are defined to measure the TOA. The principal two techniques are one-way ranging (OWR) and two-way ranging (TWR) [15]. The OWR technique (Figure 1(a)) is based on one transmission of a signal between the Tx and the Rx. This technique supposes that Tx and Rx are synchronised. The TOA is then measured as the difference between the time of reception at the Rx and the time of transmission at the Tx [15]. Unlike the OWR technique, the TWR technique (Figure 1(b)) is based on two transmissions of a signal between Tx and Rx. These two transmissions overcome the constraint of synchronisation. A short time of signal reply must be taken into consideration in this technique. This time is the time spent by the Rx to resend the received signal. The TOA is then measured as the half of the difference between the time spent by the Tx to receive the signal and the time of reply [15]. The TWR technique is adopted in the impulse radio ultra-wideband (UWB) defined in the IEEE 802.15.4a standard [16]. The large bandwidth and penetration capability of UWB signals make them suitable for ranging purposes [17]. Nevertheless, this accuracy may be reduced by multipath propagation, non-light-of-sight and multiuser interferences [18]. In wireless local area network (IEEE 802.11 standard), TOA can be estimated from round-trip time (RTT) technique. RTT is the time a signal takes to travel from a Tx to a Rx and back again. RTT is estimated by measuring the time elapsed between two consecutive frames: a link layer data frame sent by the Tx and the corresponding link layer acknowledgement sent by the Rx. T0

where k2 is the variance on the range estimation. Therefore, the pdf of rk is given by 1 e pk .rk / D p 2k

.rk dk /2 2 2 k

(11)

We assume independence between the estimated ranges. Hence, the joint pdf of the K estimated ranges is

p1;::;K .r1 ; : : : rK / D

K Y kD1

1

e p 2k

.rk dk /2 2 2 k

(12)

The log-likelihood function FToA is then given by FTOA D

K X kD1

1

.r dk /2 ln 2 k

p

2k

!

2k

(13)

O satisfies the following expression: Hence, the solution X rFTOA D

K X 1 .rk dk / O X Xk D 0 2 dk

kD1

(14)

k

2.3. Time difference of arrival The TDOA measures the difference of ranges between Tx–Rx pairs. A straightforward method of estimating the TDOA is to cross-correlate the signals arriving at a pair of nodes. In [19] and [20], authors proposed two similar techniques to estimate the TDOA for respectively IEEE 802.11 and UWB networks. In UWB, TDOA measurements may be more accurate because of the high time precision due to large bandwidth allocated to UWB communications. The T0

T1

T1

Tx

Isochronous

Tx

Rx

Rx

ToA

Tr

ToA

ToA = T1 - T0

ToA = ½[(T1 – T0) – Tr]

(a) OWR

(b) TWR

Figure 1. One-way ranging (OWR) versus two-way ranging (TWR) techniques. ToA, time of arrival; Tx, transmitter; Rx, receiver. Eur. Trans. Telecomms. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/ett



localisation by TDOA is based on the perfect synchronisation between anchor nodes. Nevertheless, synchronisation can be avoided like in the technique presented in [19]. Let us denote the true value of the difference of ranges between the couple of nodes .i ; j / as (15) dij D di dj D kX Xi k X Xj where di and dj are the ranges that separates the MS and respectively the i th and j th anchor nodes involved in the measurement of the TDOA. We assume that we have access to a measurement of dij that is centred on the true value plus a Gaussian noise ij ; let us denote this measurement: (16) ıij N dij ; ij2

2.4. Hybrid techniques In heterogeneous networks, the MS may be able to conjointly support two or more different RATs [31]. Indeed, currently developed mobile terminals are mostly able to conjointly connect to 2G, 3G, wireless local area network, bluetooth and infrared networks. This fact has its advantages in heterogeneous localisation systems. Despite the fact that RSSI measurements are usually available and very cheap to be measured, other LDPs are usually available in heterogeneous networks; thanks to different technologies supported by the new terminals like ranging and RTT. The fusion of different LDPs seems to be a promising technique of localisation. Assuming that all measurements (RSSI, TOA and TDOA) are independent, we define the hybrid estimator as

The pdf of ıij is then given by

1

pij .ıij / D p e 2ij

rFHybrid D rFRSSI C rFTOA C rFTDOA D 0

.ıij dij /2 2 2 ij

(17)

Assuming without loss of generality that all TDOAs are being measured with reference to the first anchor node (i.e. j D 1), the joint pdf assuming independence between TDOAs is then given by p2;:::;K .ı21 ; : : : ; ıK1 / D

K Y

p

kD2

1 2k1

e

.ık1 dk1 /2 2 2 k1

(18) Hence, the log-likelihood function FTDoA is then given by FTDOA D

K X

2

kD2

.dk1 dk1 / 2 2k1

ln

p

2k1

(19)

O satisfies the following explicit Thus, the ML solution X expression: ! K X O X1 O Xk X .ık1 dk1 / X rFTDOA D dk d1 2 kD2

D0

(21)

k1

(20)

Different techniques have been proposed to resolve Equations (8), (14) and (20). Linearisation techniques have been used widely like in [21–24] where different estimators are proposed for RSSI-based, TOA-based and TDOAbased localisations. The authors in [21] give a survey of principal used techniques of localisation using TOA and TDOA. These techniques may be used also for RSSI. The method based on Taylor series has been also described in this paper. Genetic algorithms have been also used to solve the problem like in [25] for robotic purposes. On the other hand, iterative optimisation techniques have been used recently in localisation problems like in [26–28]. Semidefinite and convex optimisation techniques are also used for localisation purposes like in [29, 30].

where rFRSSI , rFTOA and rFTDOA are defined respectively by Equations (9), (14) and (20). Different techniques and estimators are proposed for hybrid localisation purposes. In [32], we proposed a scheme to fuse RSSI and TOA for localisation within UWB standard. A technique for fusing RSSI coming from Wi-Fi with TDOA coming from cellular WIMAX is also proposed in [33].

3. CRAMER–RAO LOWER BOUNDS The CRLB is very useful to evaluate theoretical performances of any estimator. It expresses a lower bound on the variance of estimators of a deterministic parameter. The CRLB has been used to evaluate localisation techniques limits like in [9, 34–36]. In its simplest form, the bound states that the variance of any unbiased estimator is at least as high as the inverse of the Fisher information [37]. For different nonhybrid and hybrid localisation techniques to be evaluated, let X represents the N 1 position vector of the targeted MS (N D 2 .3/ in 2D (3D)) with elements .xi /i . The Fisher information takes the form of an N N matrix. This matrix is called the Fisher information matrix (FIM) and its typical element is defined by [38] .FIM .X//i;j D E

@ @ ln f .X/ ln f .X/ jX @xi @xj

(22)

where f is the log-likelihood function that is equal to FRSSI , FTOA and FTDOA respectively for RSSI, TOA and TDOA cases. O of X The covariance matrix of any unbiased estimator X is then bounded by the inverse of the FIM [38]: O .FIM .X//1 var X

(23)

Eur. Trans. Telecomms. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/ett



The CRLB is then defined as the trace of the inverse of the FIM [38]. o n O D t r .FIM .X//1 CRLB X

The CRLB is then computed using Equation (24), and it is given by K P

(24) 2 TOA D

These theoretic results will be then applied on RSSI, TOA and TDOA localisation to assess theoretic performances of different nonhybrid and hybrid localisation schemes. In fact, because the ML estimator is usually unbiased, the results presented in this paper are valid only asymptotically. Indeed, the ML estimator asymptotically attains the CRLB as the data length tends to infinity. Nevertheless, the main goal of this paper is to evaluate the effect of different LDPs fusion on the positioning accuracy. This comparison should be fairly carried out using these asymptotic expressions of CRLBs. 3.1. Cramer–Rao lower bounds of nonhybrid techniques The FIM for respectively RSSI, TOA and TDOA are defined by i h FIMRSSI D E rFRSSI rFRSSI T

iD1 K P iD1

.xxi /2 i2 di2

K P

.yyi /2 i2 di2

iD1

FIMTOA D E rFTOA rFTOA T

i

i h FIMTDOA D E rFTDOA rFTDOA T

K P iD1

2 D TOA 1 2

K P K P iD1 j D1

..xxi /.yyj /.xxi /.yyj //2 i2 j2 di2 dj2

2 D TOA

K P

iD1 K P

2 FIMTOA D E 4

K X 1 .ri di /2

4 iD1 i

di

2

3

Developing this expression leads to

FIMTOA D

T K X .X Xi / X Xj iD1

FIMTOA D

K X iD1

2 1 6 4 i2

2 D RSSI

iD1 K P K P

i2

.xxi /2 di 2 .xxi /.yyi / di 2

sin2 .'j i / 2i2 j2

(33)

.1CSi2 / Si2 di2

.1CSi2 /.1CSj2 / sin2 .'j i /

(34)

2Si2 Sj2 di2 dj2

iD1 j D1

K P 2 TDOA D

T .X Xi / X Xj 5 (28)

1 i2

Similarly to Equation (33), the CRLBs of RSSI and TDOA cases are given respectively by K P

Because calculations are similar, we will only present the case of TOA. Using the expression of rFTOA defined in Equation (14) and assuming independence between TOA measurements, the FIM given by Equation (26) can be written as

1 i2

(32) Let 'i be the angle between the distance di and the line (y D 0). We obtain cos.'i / D .x xi /=di and sin.'i / D .y yi /di . Introducing that in Equation (32), using the trigonometric equality cos.'i / sin.'j /cos.'i / sin.'j / D sin.'j 'i /, and defining 'j i D 'j 'i , the angle between di and dj lead to the simplified expression of the CRLB:

(25)

(27)

!2 .xxi /.yyi / i2 di2

(31) After simplifications, Equation (31) can be written as

K P

(26)

K P iD1

iD1 j D1

h

1 i2

1cos.'i 1 / i21

iD2 K P K P

iD2 j D2

sin2 .'j i / i21 j21

(35)

For given Sk D S , k D and i1 D 1 , Equations (34), (33) and (35) become, respectively, 2 D TOA

(29)

2K 2 K P K P sin2 .'j i /

(36)

iD1 j D1

.xxi /.yyi / di 2 .yyi /2 di 2

3 7 5 (30)


2 RSSI

2S 2 1CS 2

D

K P K P iD1 j D1

K P iD1

1 di2

sin2 .'j i / di2 dj2

(37)


2 TDOA D

12

K P


.1 cos.'i1 //

iD2

(38)

K P K P

sin2 .'j i /

JHDF D JRSSI C JTOA C JTDOA

iD2 j D2

Hence, we obtain three purely geometric quantities respectively for RSSI, TOA and TDOA as follows: 1

2 gTOA D

K P K P

(39)

sin2 .'j i /

iD1 j D1

K P 2 gRSSI

D

iD1 K P

K P

iD1 j D1

K P

(40)

sin2 .'j i / di2 dj2

.1 cos.'i1 //

iD2 K P K P

2 gTDOA D

1 di2

(41) sin2 .'

ji/

iD2 j D2

These quantities define the geometric dilution of precision (GDOP) for respectively RSSI, TOA and TDOA. Similarly and with the utilisation of Equations (44) and (45), the GDOP for hybrid schemes can be obtained. The GDOP is commonly defined as the term associated with errors in position caused by the relative location of anchor nodes with which LDP measurements are performed [39–41]. Moreover, the GDOP matrix is defined as the unweighted FIM [42]. This GDOP matrix is given, for respectively RSSI, TOA and TDOA, by 2 K X 6 GRSSI D GTOA D 4 iD1

.xxi /2 di 4 .xxi /.yyi / di 4

in the scenario. That is, for a scenario that implies RSSI, TOA and TDOA:

.xxi /.yyi / di 4 .yyi /2 di 4

3 7 5

(44)

Consequently, the CRLB is given by o n 2 D t r .JHDF /1 HDF

(45)

These CRLBs give a theoretical prior information about the best localisation accuracy that can be achieved for a given localisation problem. These expressions of different CRLBs show that the localisation accuracy depends on different parameters: The nature of used LDPs (RSSI, TOA or TDOA): for

each LDP or combination of LDPs, a different expression of the CRLB is defined. Hence, the achieved localisation accuracy depends strongly on the implied LDP (or combination of LDPs). The number of used LDP: depending on the values of K, the localisation accuracy may increase or decrease. Nevertheless, using more LDPs does not mean automatically a better localisation accuracy. For example, four cellular RSSIs may not perform better than three UWB-based TOAs. In fact, this depends mainly on the two next parameters. The precision of the used LDP: given by sh i , i and i1 for respectively RSSI, TOA and TDOA. Their values depend on different parameters (RAT, radio channel, propagation phenomena, etc). The position of the targeted device with respect to anchor nodes: that is, the relative geometry of the localisation problem. The position of an additional anchor node must be properly chosen to enhance the localisation accuracy. In fact, this parameter is represented in the CRLB expressions by sin.'j i /. This is very interesting when the system has to choose an additional LDP to enhance the localisation accuracy. It must choose the LDP that minimises the CRLB.

(42)

GTDOA D

K X

2 4

iD2

2 xxi 1 xx di d1 xxi yyi yy1 xx1 di d1 di d1

xxi di

1 xx d 1

yyi di

yyi 1 yy di d1 2 yy1 d1

3 5

(43) 3.2. Cramer–Rao lower bounds of hybrid techniques In the case of heterogeneous scenario and assuming independence between all measurements, the FIM can be defined as the sum of the FIMs of different LDP implied

In hybrid cases, different fusion schemes can be studied, namely RSSI C TOA, RSSI C TDOA, TOA C TDOA and RSSI C TOA C TDOA. In the next section, the considered nonhybrid and hybrid schemes will be evaluated and compared by means of Monte Carlo simulation while considering realistic LDPs.

4. SIMULATIONS AND DISCUSSIONS 4.1. Simulation scenario and parameters The assumed scenario is a situation where the targeted mobile is connected to different anchors from which it is able to have different LDPs. Let K be the total number of all anchors implied in the scenario. Without any loss of Eur. Trans. Telecomms. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/ett


generality, assume that the targeted MS can have p RSSIs (Pk ) from anchors with indexes k 2 .1; : : : ; p/, q p TOAs (k ) from anchors with indexes k 2 .p C 1; : : : ; q/ and K q1 TDOAs (k.qC1/ ) from anchors with indexes k 2 .q C2; : : : ; K/ obtained with reference to the .q C1/th anchor. Figure 2 depicts an example of hybrid scenario where p D 4, q D 8 and K D 12. The area is a square of L L m2 . L is fixed to 20 m unless otherwise noted. This scenario will be used for all simulations. The statistical models used in simulations are extracted from an UWB measurement campaign [43] carried out within the FP7 European project WHERE. A summary of these models is given in Table I: 4.2. Geometric distribution of Cramer–Rao lower bounds for both nonhybrid and hybrid localisation schemes The CRLBs for RSSI, TOA and TDOA are shown respectively in Figure 3 (a), (b) and (c). These figures show the geometric distribution of the CRLB over the assumed area. The comparison between these three figures obviously reveals that time-based techniques (i.e. TOA and TDOA) have better overall localisation accuracy than power-based technique (RSSI). In addition, comparison between Figure 3(b) and (c) shows that the TOA technique outperforms the TDOA technique. Moreover, these three figures show that the CRLB depends on the position of


Table I. Statistical models used for simulations. RSSI

P0 D 36:03 dBm , np D 2:38, d0 D 1 m sh k D 3:98 dBm

TOA TDOA

k D 1:142 m k .q C1/ D 1:85 m

RSSI, received signal strength indicator; TOA, time of arrival; TDOA, time difference of arrival.

targeted MS with respect to the configuration of anchor nodes. This dependency is quite different from one LDP to another. In the RSSI case, the localisation accuracy is degraded as the MS approaches an anchor node or an edge of the considered squared area. In the case of TOA, the difficulties of localisation are located around the anchor nodes, and the localisation accuracy is as better as the MS moves toward the centre of the area (i.e. equidistant to all anchors). In the TDOA case, the difficulties of localisation are mainly located around the edges of the assumed squared area. To theoretically evaluate the effect of fusion of different LDP on localisation accuracy, we plot in Figure 4(a)–(d) the values of CRLBs over the simulated area for respectively (RSSICTOA), (RSSICTDOA), (TOACTDOA) and (RSSICTOACTDOA) hybrid schemes. These figures show that the overall localisation accuracy is enhanced when fusing LDPs. The comparison between these figures reveals that the fusion of the three LDPs is the scheme that offers the best accuracy. The fusion of time-based LDPs

Figure 2. Heterogeneous generic scenario. RSS, received signal strength; ToA, time of arrival; TDoA, time difference of arrival; BS, base station; MS, mobile station. Eur. Trans. Telecomms. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/ett


(a) RSSI


(b) TOA

(c) TDOA Figure 3. Cramer–Rao lower bound values over the L L area for nonhybrid techniques. RSSI, received signal strength indicator; TOA, time of arrival; TDOA, time difference of arrival; CRLB, Cramer–Rao lower bound.

(TOA and TDOA) themselves offers a better accuracy than the two schemes (RSSICTOA) and (RSSICTDOA) that offer close performances. Table II gives the average ACRLB root-squared values (denoted ACRLB) for each localisation scheme. The comparison between these different values of ACRLB reveals that the accuracy enhancement is more drastic when time-based LDPs are added to RSSI. By contrast, adding RSSI, which are usually imprecise to TOA or TDOA, is not very interesting if these time-based LDPs are very precise. 4.3. Effect of location-dependent parameter accuracies 4.3.1. Effect of the RSSI shadowing. For each hybrid scheme that involve RSSI, Figure 5 plots the evolution of ACRLB over the area (L D 20 m) with respect to the standard deviation of RSSI shadowing (sh k ). Obviously, when sh k increases the accuracy of RSSI-based localisation scheme decreases. Nevertheless, when fusing RSSI with other time-based LDPs, the effect of shadowing (i.e. sh k ) is attenuated. For sh k D 6 dBnJ , a gain of 5:3 and 4:9 m is performed when respectively

adding TOA and TDOA to RSSI. This gain reaches 5:6 m when fusing both TOA and TDOA with RSSI. 4.3.2. Effect of the time of arrival ranging error. Figure 6 plots the evolution of ACRLB with respect to the TOA ranging error (k ) for techniques involving TOA. The figure shows obviously that the accuracy of nonhybrid TOA localisation scheme deteriorates as k increases. The adding of RSSI measurements enhances this accuracy and reduce the effect of ranging error. This enhancement provided by RSSI is as important as the TOA is less precise (a gain of 0:21 m at k D 2 m and 2:9 m at k D 6 m). Moreover, the curves of both (TOACTDOA) and (RSSICTOACTDOA) techniques reveal that fusing TOA with TDOA, or both TDOA and RSSI drastically attenuate the effect of TOA ranging error on localisation accuracy. When adding RSSI and TDOA, a gain of 1:1 m is performed for k D 2 m. 4.3.3. Effect of the time difference of arrival ranging error. The effects of TDOA ranging error (k.qC1/ ) on localisation techniques involving TDOA are plotted in Figure 7. Eur. Trans. Telecomms. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/ett



(a) RSSI + TOA

(b) RSSI + TDOA

(c) TOA + TDOA

(d) RSSI + TOA + TDOA

Figure 4. Cumulative density functions of localisation error using different estimators applied on the fusion of received signal strength indicator (RSSI), time of arrival (TOA), and time difference of arrival (TDOA). CRLB, Cramer–Rao lower bound.

Table II. ACRLB values over the L L area for nonhybrid and hybrid schemes. LDP

4.4. Choice of localisation techniques based on Cramer–Rao lower bound

ACRLB (m)

RSSI TDOA TOA RSSICTDOA RSSICTOA TOACTDOA RSSICTOACTDOA

4:378 1:600 1:165 1:334 1:065 0:875 0:824

ACRLB, asymptotic Cramer–Rao lower bound; LDP, locationdependent parameter; RSSI, received signal strength indicator; TDOA, time difference of arrival; TOA, time of arrival.

This figure reveals similar remarks like in the TOA and RSSI cases. Indeed, as k.qC1/ increases, the accuracy of TDOA technique deteriorates. The RSSI reduces the ACRLB by 0:3 m at k.qC1/ D 2 m. But adding TOA or both RSSI and TOA enhances deeply the localisation accuracy and compensates the effect of TDOA ranging error. Eur. Trans. Telecomms. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/ett

The theoretical and simulated results presented in Sections 3 and 4 emphasise the fact that CRLB can be used as a criterion for the choice of the best localisation scheme (i.e. LDPs) and the planning of localisation systems. Because of its theoretical nature, the CRLB can be calculated before performing localisation task. Hence, it can offer a prior information about the accuracy level reachable by the system using the available resources (anchor nodes, measurements techniques, etc.) while considering accuracies of performed measurements. The CRLB can be presented as a criterion for choosing additional (LDPs) to enhance the actual localisation accuracy of the system. This is shown by the different steps in Figure 8. Because localisation systems cannot be disassociated from communication systems already installed in different area, the available number of anchor nodes may be known and fixed. In this case, the accuracy of localisation technique depends only on the nature (RSSI, TOA or TDOA), the number and the accuracies of performed measurements. We are mainly interested in the scenario where


Figure 5. Asymptotic Cramer–Rao lower bound (CRLB) of different techniques with respect to the received signal strength indicator (RSSI) shadowing. TOA, time of arrival; TDOA, time difference of arrival.


Figure 7. Asymptotic Cramer–Rao lower bound (CRLB) of different techniques with respect to time difference of arrival (TDOA) ranging error. RSSI, received signal strength indicator; TOA, time of arrival.

Positioning with available LDPs

Sensing of new possible LDPs

Compute the different CRLBs Figure 6. Asymptotic Cramer–Rao lower bound (CRLB) of different techniques with respect to time of arrival (TOA) ranging error. RSSI, received signal strength indicator; TDOA, time difference of arrival.

the MS is using all available RSSI measurements and is trying to add some TOAs or TDOAs to reach the requested accuracy. Here, we assume that the targeted MS communicates first to discover all available ranging-capable devices. Then the MS uses CRLB, computed using available estimation of its position, to choose the best set of these devices to perform ranging with them, and then the MS performs the localisation task. This scenario is justified by the following rational facts: (1) RSSI measurements are usually available with no additional costs in all RATs. (2) TOA and TDOA measurements are more precise than RSSI especially in UWB standard. (3) TOAs and TDOAs are measured through ranging techniques that require additional resources and costs.

Choose the best LDPs in the CRLBsense

Figure 8. The Cramer–Rao lower bound (CRLB) as a criterion for choosing additional location-dependent parameters (LDPs).

(4) These ranging procedures may cause network congestion and reduce network throughput. Hence, ranging attempts must be reduced as much as possible. The steps presented in Figure 8 are applied on respectively the two hybrid schemes (4 RSSI C 2 TOA) and (4 RSSI C 2 TDOA). We assume the targeted MS has Eur. Trans. Telecomms. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/ett



(a)

(b)

Figure 9. Cumulative density function of localisation error for the hybrid scheme (4 RSSI C 2 TOA (TDOA)) with TOA (TDOA) chosen first randomly and then based on Cramer–Rao lower bound (CRLB). RSSI, received signal strength indicator; TOA, time of arrival; TDOA, time difference of arrival.

access to all available RSSIs (four in the assumed scenario) and seeks to enhance its localisation accuracy using two additional TOAs or TDOAs. These additional measurements can be taken randomly or using the CRLB as a criterion. Figure 9(a) and (b) plots the cumulative density functions (CDFs) of localisation errors for both randombased and CRLB-based cases. In the first case, the two additional time-based LDPs are chosen randomly from the possible LDPs (see Figure 2). The second scenario chooses the couple of additional LDPs on the basis of the calculation of CRLB of all possible couples (in the assumed scenario: six couples of TOAs and three couples of TDOAs). At each iteration, the couple that offers the lowest CRLB value is chosen, and localisation is performed by ML technique using the four available RSSIs and the two chosen TOAs (respectively TDOAs). Figure 9 shows that the second scenario outperforms the first one for both (RSSICTOA) and (RSSICTDOA) schemes. This result justifies the hypothesis that CRLB can be used as a criteria to enhance localisation accuracy. Nevertheless, this approach must be validated by more realistic simulations or by implementing it on hardware to evaluate the cost of CRLB computing.

5. CONCLUSION In this paper, we have investigated the theoretic performances of nonhybrid and hybrid localisation techniques based on RSSI, TOA and TDOA. To evaluate the performances of these techniques, we derived ACRLB for the different studied schemes. The assumption of independence between LDPs made the deriving of hybrid CRLBs easier and simpler. We have carried out simulations to support our theoretic results. Simulations showed that localisation accuracy depends mainly on the number, the nature, the precision of the measurements and the geometric characteristics of the problem (i.e the size of the area and the position of targeted node with respect to anchor nodes Eur. Trans. Telecomms. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/ett

positions). Hence, a localisation system must be conceived smartly to choose the best geometric configuration and measurements to offer the position accuracy requested by the application and/or the user. The CRLB is one of the most interesting criteria to be respected before conceiving and implementing a localisation system. The next step will be to re-evaluate CRLBs for correlated and/or non-Gaussian assumptions and also to consider biased measurements.

ACKNOWLEDGEMENTS This work has been performed in the framework of the FP7 project ICT-248894 WHERE2 (Wireless Hybrid Enhanced Mobile Radio Estimators—Phase 2) that is partly funded by the European Union.

REFERENCES 1. Schiller JH, Voisard A. Location-based services. Morgan Kaufmann Publishers: California, 2004. 2. Simic MI, Pejovic PV. A comparison of three methods to determine mobile station location in cellular communication systems. European Transactions on Telecommunications 2009; 20: 711–721. DOI: 10.1002/ett.1333. 3. Frattasi S, Monti M. Ad-coop positioning system (ACPS): positioning for cooperative users in hybrid cellular ad-hoc networks. European Transactions on Telecommunications 2008; 19: 923–934. DOI: 10.1002/ett.1223. 4. Barcelo-Arroyo F, Martin-Escalona I, Manente C. A field study on the fusion of terrestrial and satellite location methods in urban cellular networks. European Transactions on Telecommunications 2010; 21: 632–639. DOI: 10.1002/ett.1433.


5. Chang C, Sahai A. Cramér–Rao-type bounds for localization. EURASIP Journal on Applied Signal Processing 2006: 2006. 6. Fritsche C, Klein A. Cramér–Rao lower bounds for hybrid localization of mobile terminals, In Proceedings of 5th Workshop on Positioning, Navigation, and Communication, WPNC, 2008; 157–164. 7. Patwari N, et al. Relative location estimation in wireless sensor networks. IEEE Transactions on Signal Processing 2003; 51(8): 2137–2148. 8. Yu K. 3-D localization error analysis in wireless networks. IEEE Transactions on Wireless Communications 2007; 6(10): 3472–3481. 9. Catovic A., Sahinoglu Z. The Cramer–Rao bounds of hybrid TOA/RSS and TDOA/RSS location estimation schemes. IEEE Communications Letters 2004; 8(10): 626–628. 10. Figel WG, Shepherd NH, Trammel WF. Vehicle location by a signal attenuation method. IEEE Transactions on Vehicular Technology 1969; 18(3): 245–251. 11. Laitinen H, Juurakko S, Lahti T, Korhonen R, Lahteenmaki J. Experimental evaluation of location methods based on signal-strength measurements. IEEE Transactions on Vehicular Technology 2007; 56(1): 287–296. 12. Rappaport TS. Wireless Communications: Principles and Practice, Second Edition. Prentice Hall: New Jersey, 2002. 13. Goldsmith A. Wireless Communications. Cambridge University Press: Cambridge, 2005. 14. Marvin K. Probability Distributions Involving Gaussian Random Variables: A Handbook for Engineers and Scientists, Springer International Series in Engineering and Computer Science. Springer: Berlin, 2006. 15. Dardari D, Conti A, Ferner UJ, Giorgetti A, Win MZ. Ranging with ultrawide bandwidth signals in multipath environments. Proceedings of IEEE 2009; 97(2): 404–426, Special issue on Ultra-Wide Bandwidth (UWB) Technology & Emerging Applications, Invited Paper. 16. IEEE 802.15.4a/D4 (Amendment of IEEE Std 802. 15.4), Part 15.4: Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications for Low-Rate Wireless Personal Area Networks (LRWPANs), 2006. 17. Sahinoglu Z, Gezici S, Guvenc I. Ultra-Wideband Positioning Systems: Theoretical Limits, Ranging Algorithms, and Protocols. Cambridge University Press: Cambridge, 2008. 18. Gezici S, Poor HV. Position estimation via ultrawideband signals. Proceedings of the IEEE (Special Issue on UWB Technology and Emerging Applications) 2009; 97(2): 386–403.


19. Ylianttila M, Li X, Pahlavan K, Latva-aho M. Comparison of indoor geolocation methods in DSSS and OFDM wireless LAN systems. Proceedings of IEEE VTS-Fall VTC 2000 2000; 6: 3015–3020. 20. Yan B, Xiao-chun L, Jing-song X, Jin W. Research on UWB indoor positioning based on TDOA technique, In Proceedings of IEEE International Conference on Electronic Measurement & Instruments (ICEMI’09), 2009; 167–170. 21. Shen G, Zetik R, Thoma R. Performance comparison of TOA and TDOA based location estimation algorithms in LOS environment, In Proceedings of the Workshop On Positioning, Navigation And Communication 2008, 2008. 22. Cheung K, So H, Ma W, Chan Y. A constrained least squares approach to mobile positioning: algorithms and optimality. EURASIP Journal on Applied Signal Processing 2006; 17: 150–150. 23. Laaraiedh M, Avrillon S, Uguen B. Enhancing positioning accuracy through RSS based ranging and weighted least square approximation, In Proceedings of POCA conference, 2009. 24. Laaraiedh M, Avrillon S, Uguen B. Overcoming singularities in TDoA based location estimation using total least square, In Proceedings of IEEE International Conference on Signals, Circuits & Systems (SCS09), 2009. 25. Armingol JM, Moreno LE, de la Escalera A, Salichs MA. A genetic algorithm for mobile robot localization using ultrasonic sensors. Journal of Intelligent and Robotic Systems 2002; 34(2): 135–154. 26. Laaraiedh M, Avrillon S, Uguen B. A maximum likelihood TOA based estimator for localization in heterogeneous networks, International Journal on Communication. International Journal of Communications, Network and Systems Sciences (IJCNS) 2010; 3(1): 38–42. 27. Beck A, Teboulle M, Chikishev Z. Iterative minimization schemes for solving the single source localization problem. SIAM Journal on Optimization 2008; 19(3): 1397–1416. 28. Laaraiedh M, Avrillon S, Uguen B. Enhancing positioning accuracy through direct position estimators based on hybrid RSS data fusion, In Proceedings of IEEE VTC Spring Conference, 2009. 29. Carter M, Jin H, Saunders M, Ye Y. SPASELOC: an adaptive subproblem algorithm for scalable wireless sensor network localization. SIAM Journal on Optimization 2006; 17: 1102–1128. 30. Cheung KW, Ma WK, So HC. Accurate approximation algorithm for TOA-based maximum likelihood mobile location using semidefinite programming. Proceedings of International Conference on Acoustic, Speech, and Signal Processing 2004; 2: 145–148. Eur. Trans. Telecomms. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/ett


31. Sauter M. Beyond 3G–-Bringing Networks, Terminals and the Web Together: LTE, WiMAX, IMS, 4G Devices and the Mobile Web 2.0. Wiley: Hoboken, 2009. 32. Laaraiedh M, Avrillon S, Uguen B. Hybrid Data fusion techniques for localization in UWB networks, In Proceedings of Workshop on Positioning Navigation and Communication 2009, 2009. 33. Mayorga CLF, Rosa FD, Simone SAWG, Raynal MCN, Figueiras J, Frattasi S. Cooperative positioning techniques for mobile localization in 4G cellular networks, In Proceedings of IEEE International Conference on Pervasive Services (ICPS07), 2007. 34. Shen Y, Win MZ. Fundamental limits of wideband localization–-part I: a general framework. IEEE Transactions on Information Theory 2010; 56(10): 4956–4980. 35. Shen Y, Win MZ. Fundamental limits of wideband localization—part II: cooperative networks. IEEE Transactions on Information Theory 2010; 56(10): 4981–5000. 36. Jourdan D, Dardari D, Win M. Position error bound for UWB localization in dense cluttered environments. IEEE Transactions on Aerospace and Electronic Systems 2008; 44(2): 613–628. 37. Steven M. Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice-Hall: New Jersey, 1993. 38. Shao J. Mathematical Statistics, 2nd edition. Springer: Berlin, 2007. 39. Frenkel G. Geometric dilution of position (GDOP) in position determination through radio signals. Proceedings of the IEEE 1973; 61(4): 496–497. 40. Xiwu L, Kaihua L, Po H. Geometry influence on GDOP in TOA and AOA positioning systems. Proceeding of the 2010 Second International Conference on Networks Security Wireless Communications and Trusted Computing (NSWCTC) 2010; 2: 58–61. 41. Sharp I, Kegen Y, Guo YJ. GDOP analysis for positioning system design. IEEE Transactions on Vehicular Technology 2009; 58(7): 3371–3382. 42. Chaffee J, Abel J. GDOP and the Cramer–Rao bound, In Proceedings of IEEE Position Location and Navigation Symposium, 2002; 663–668.



43. ICT-217033 WHERE Project, Deliverable 4.1: measurements of location-dependent channel features, 2008.

AUTHORS’ BIOGRAPHIES Mohamed Laaraiedh received his engineering degree in wireless networks and services from higher school of telecommunications of Tunis in Tunisia (SupCom) in 2007 and his PhD degree in signal processing and telecommunications from the University of Rennes 1 in France. He is currently a PhD and an associate researcher at the Institute for Electronics and Telecommunication of Rennes Labs in the University of Rennes 1. He is mainly interested in localisation techniques within wireless networks. Stéphane Avrillon received his PhD in 2004 on ‘Optimization of communication terminal performances by mastering of the filtering distribution in a RF transmitter’. He is currently an associate professor at the University of Rennes 1. He is a member of the communication and propagation team of the Institute for Electronics and Telecommunication of Rennes. His main interest goes from antenna design characterisation and modelling, diversity antennas, multiple-input multiple-output channel characterisation, ultra-wideband systems, and recently, to indoor positioning radio system. Bernard Uguen received his Dipl-Ing and his PhD degree in Electronics and Signal Processing from the National Institute of Applied Sciences (INSA) of Rennes, France, in 1988 and 1992, respectively. From 1993 to 2006, he has been an associate professor at the Electronics and Communications Systems Department of the INSA of Rennes. In September 2006, he joined the University of Rennes 1 where he is currently a full professor. He belongs, as a researcher, to the Rennes institute for Electronics and Telecommunications. His main research interests are electromagnetic and diffraction theories, propagation modelling, radar and ultra-wideband (UWB) technology. He has been involved in the application of 3D ray tracing for analysis of UWB localisation systems in multipath channel. His current interests are related to UWB communications and localisation within wireless network.