TDOA positioning in NLOS scenarios by particle filtering

Mauro Boccadoro • Guido De Angelis •

Paolo Valigi

Published online: 23 February 2012

� Springer Science+Business Media, LLC 2012

Abstract A method is proposed for position estimation

from non line of sight time difference of arrivals (TDOA)

measurements. A general measurement model for TDOA

accounting for non line of sight conditions is developed;

then, several simplifying working assumptions regarding

this model are discussed to allow the efficient implemen-

tation of a particle filter localization algorithm. This algo-

rithm is tested and compared with an extended Kalman

filter procedure, both in simulation, generating artificial

measures, and with real data.

Keywords Mobile location � Time difference of arrival

(TDOA) � Particle filter � Non line of sight (NLOS)

1 Introduction

The wireless positioning problem, using existing 2 and 3G

cellular network infrastructure, has been addressed in past

years in response to an essential public safety feature of

cellular systems. This problem has been thoroughly

investigated in past years; surveys of the various tech-

niques for wireless positioning can be found in [1, 21].

In many cases the measurements acquired are the time-

difference-of-arrival (TDOA) of the pilot channel codes of

base stations (BS) in visibility to the mobile station (MS).

The measurement models in general contemplate a ‘‘stan-

dard system measurement noise’’, under line-of-sight

(LOS) conditions, usually modeled by a zero-mean

Gaussian distribution, and an additional error, independent

from the first, in case of non-LOS (NLOS) condition, due

to multipath propagation. NLOS errors are typically rec-

ognized preponderant with respect to LOS errors and hence

the main source of performance degradation of positioning

algorithms (see e.g., [15, 26]).

Various approaches are proposed to address this issue.

Several contributions, e.g., [4, 6, 15, 27], rely on methods

to discriminate between LOS and NLOS readings, and

consequently weigh less, or neglect, the measurement

which are recognized as NLOS; in particular, in [11],

NLOS errors are estimated and then subtracted from the

measured distance before trilateration is performed. Ref.

[14] proposed a Markov chain process to model the switch

between LOS and NLOS conditions, and accordingly set

two different range measurement equations. There, the

position estimation, for TOA measurements, relies on two

interacting Kalman Filters which are relative to the two

possible sight conditions. Alternatively, exploiting the

geometrical constraints characterizing the problem, the

measurements possibly perceived under NLOS conditions

are adjusted [25]; another method, principally based on

geometric properties, is that presented in [18].

In general, both the position of the MS and the LOS/

NLOS condition (typically modeled by a two-state Markov

process) constitute the state to be estimated; this approach

is pursued e.g., in [5, 13, 19]. In [17] five different geo-

location algorithms have been compared, based on the

Hough transform, particle filtering and Bayesian random-

ized estimator.

Particle filtering is exploited for positioning problems

also in [12, 20], although detailed measurement models of

M. Boccadoro � G. De Angelis � P. Valigi (&)

Department of Electronic and Information Engineering (DIEI),

University of Perugia, 06125 Perugia, Italy

e-mail: valigi@diei.unipg.it

M. Boccadoro

e-mail: boccadoro@diei.unipg.it

G. De Angelis

e-mail: guidodeangelis@ieee.org

123

Wireless Netw (2012) 18:579–589

DOI 10.1007/s11276-012-0420-9

multiple TDOA readings are not provided. This approach,

also adopted here, is presented, e.g., in [23], and is sur-

veyed in [24], which addresses a localization problem

(although considering different observation data than

TDOA). In [24] the particle filter (PF) is referred to as

‘‘sequential Monte Carlo’’ method; making reference to

this work, the main ingredients needed to set up a PF

algorithm are the state transition equation (Eq. 3 of [24])

and a measurement model (Eq. 4 of [24], there referred to

as observation equation).

As the state transition equation is quite standard in the

localization literature, in this work we especially focus, in

Sect. 2.1, on the development of the measurement model

for multiple TDOA readings which accounts for the cor-

relation existing among the measurements due to the

common reference to a BS. The problems related to this

issue are known, although not always recognized (see [9]),

however, to our knowledge, such explicit models have not

been presented in the literature. In order to be able to

exploit in practical implementations this general model,

some simplifying hypotheses are discussed, in Sect. 2.2.

Finally, the PF positioning algorithm based on the model

developed is presented and tested in Sect. 3.

2 Problem formulation

We consider a 2-D localization problem of a MS. Denote

its actual position by p� 2 R2, and by di

* the actual distance

from the ith BS (BSi). TDOA measurements, as well as

time of arrival (TOA) measurements, are characterized by

a system measurement noise and also by a NLOS error, due

to multipath propagation (see Fig. 1). The former is typi-

cally modeled by a zero mean Gaussian distribution,

whereas regarding NLOS errors several hypothesis have

been conjectured: Gaussian [17], uniform [4], exponential

[10, 11, 16, 19]; and also Rayleigh, Rician and chi-square

[11] distributions are considered. Detailed studies on

modeling the delay spread are presented in [8]. To achieve

a probabilistic characterization of NLOS errors, actual

TOA measurements errors were reported in [22], and,

considering indoor environments, some models were

developed in, e.g., [2], supporting the hypothesis of an

exponential model for NLOS measurements.

In various works, e.g., [5, 19], a Markov Chain is

employed to model the switch between LOS and NLOS

conditions so that a TOA reading taking place at the t-th

sampling instant is

siðtÞ ¼1

cd�i ðtÞ þ wlosðtÞ þ wnlosðtÞsiðtÞ ð1Þ

where c is the speed of signal propagation, wlos and wnlos

the system measurement noise and NLOS noise, and

si 2 f0; 1g is the state of a Markov Chain modeling the

dynamics of the sight conditions between MS and each BS,

so that si(t) = 0 (si(t) = 1) if the MS is in LOS (NLOS)

condition with BSi at sampling instant t.

2.1 TDOA measurement model with correlated

readings

To derive the TDOA measurement model, denote by di the

multipath distance between the MS and BSi; then, consis-

tently with (1), assume that

diðtÞ ¼ d�i ðtÞ þ cwnlosðtÞsiðtÞ: ð2Þ

An illustration of this scenario is in Fig. 1. The pdf of

P(di|p*, si), or equivalently of P(di|di*, si), will be denoted,

according to any of the above discussed assumption

regarding wnlos, by fi;siðdiÞ. Now, in the LOS case,

(si = 0), fi,0(di) degenerates into a Dirac delta centered in

di*. Regarding the NLOS case (si = 1), one common

assumption, mentioned above, contemplates exponential

distributions for wnlos, which implies that:

fi;1ðdiÞ ¼ kie�kiðdi�d�i Þd�1ðdi � d�i Þ; ð3Þ

where d-1 is the step function (d-1(x) = 0 if x \ 0 and

d-1(x) = 1 if x C 0). We will denote for short the above

distribution by Eð1=kiÞ.

Remark 1 To our knowledge, the characterization of

NLOS errors distribution, based on true measurements, is

still an open problem. Ref. [22] is a rare example of work

reporting experimental data (Fig. 5 in [22]), however this

data were not discriminated on the basis of the true

−500 0 500 1000 1500 2000 2500 3000

0

200

400

600

800

1000

1200

1400

1600 BS1

BS2

BS3

MS

d1*

d1

d2=d

2*

d3=d

3*

Fig. 1 Illustration of measurements taken in LOS/NLOS conditions.

BS2 and BS3 are in sight of the MS (LOS condition) hence, referring

to Eq. 2, d2 = d2*, d3 = d3

*. BS1 is not visible due to the presence of an

obstacle (NLOS condition) hence the multipath distance d1 differs

from d1* (the true distance between MS and BS1)

580 Wireless Netw (2012) 18:579–589

123

distances from the BSs; in other words, following our

notation, such measurements have to be considered sam-

ples of P([di - di*] ? wlos) instead of the conditional

probability P([di - di*] ? wlos|di

*), through which an

observation equation is defined (as a consequence, if a

characterization of the pdf of the multipath error were

pursued on the basis of this data, it would be implicitly

assumed that the multipath lag is independent on the LOS

distance).

We also argue that in order to achieve a more realistic

model of NLOS errors, the autocorrelation of the signal

wnlosð�Þ should be taken into account;1 this will not be

addressed here.

In this work, we refer to a IS-95 CDMA network, where

all BSs use the same carrier frequency for the pilot channel,

and are synchronized to a common CDMA system time,

which is derived from a precise clock reference supplied by

GPS. TDOA (absolute) values can be estimated using a

well known cross-correlation based technique, by looking

for a peak in the PN code acquisition matrix [7, 27]. In this

case TDOA readings are constituted by the terms |di - dj|,

assuming no prior information available about which

multipath distance is shorter, corrupted by system mea-

surement noise, that is:

zij ¼ zji ¼ jdi � djj þ wij: ð4Þ

In case of n - 1 TDOA measurements, n BS, with BS1

the common reference, denote the readings by Z1 ¼fz21; . . .; zn1g, the sight condition by s ¼ ½s1; . . .; sn�2 f0; 1gn

, define by Fn-1(s) the pdf of P(Z1|p*,s) when

the various wij are neglected. For a single measurement,

hence 2 BS:

F1ðsÞ ¼Z1

0

f1;s1ðyÞf2;s2

ðyþ z21ÞdyþZ1

0

f1;s1ðy

þ z21Þf2;s2ðyÞdy; ð5Þ

where the two integrals arise considering the sign of d1 - d2.

For multiple readings, by the correlation existing among

the measurements due to the common reference base,

obviously PðZ1jp�Þ 6¼Q

i Pðzi1jp�Þ (see e.g., [9]), and a

computation similar to (5) should be followed, in this case

accounting for the 2n-1 combinations of signs of the

absolute values, thus computing 2n-1 integrals. For n = 3

we have (omitting the reference to the si):

F2ðsÞ ¼Z1

0

f1ðyÞf2ðyþ z21Þf3ðyþ z31Þdy

þZ1

0

f1ðyþ z31Þf2ðyþ z21 þ z31Þf3ðyÞdy

þZ1

0

f1ðyþ z21Þf2ðyÞf3ðyþ z21 þ z31Þdy

þZ1

0

f1ðyþ z21Þf2ðyÞf3ðyþ z21 � z31Þdy:

ð6Þ

The four terms in the above equation are relative to the fol-

lowing combinations of the sign of d2 - d1 and d3 � d1 :hþ;þi; hþ;�i; h�;þi; h�;�i (hence, in particular, the

first integral is relative to the case |d2 - d1| = d2 - d1

and |d3 - d1| = d3 - d1; notice that the fourth term

could be equivalently expressed asR1

0f1ðyþ z31Þf2ðyþ

z31 � z21Þf3ðyÞdy).

The pdf’s Fn-1 can be particularized according to any

standing assumption regarding NLOS error modeling, and

also considering all the possible (2n) sight conditions. This

would lead to the computation of 2n-12n integrals. Now,

considering also system measurement noise, i.e., the vari-

ous terms wij, we have that P(Z1|p*, s) results from the

convolution of Fn-1(s) with the joint distribution of the wij

(for which we may have, e.g., a multivariate Gaussian

distribution with zero mean and diagonal covariance

matrix, independently by state s). To proceed further,

several working assumptions can be made to tackle the

complexity of this model, both due to the exponential

complexity of the combination of the signs, and due to the

need to perform the convolution between Fn-1 and the pdf

characterizing the wij.

2.2 Approximated measurement model

First, if one considers only position estimation, letting

LOS/NLOS detection, as in this work, marginalization over

the variable s is not mandatory, and one could adopt a

measurement model relative to the worst case scenario,

namely si = 1, V i (denoted s = 1). In this case, the pre-

dominant contribution in P(Z1|p*, 1), in terms of variance,

is that relative to the terms |di - dj|; thus, neglecting wij we

stand with a slightly less accurate but simpler measurement

model, P(Z1|p*, 1) % Fn-1(1), (see the comparison for a

single TDOA reading in Fig. 2). It is notable, however, that

considering a gaussian additive term wij to |di - dj| could

be not satisfactory as this makes non zero the probability to

get negative readings, which is not realistic, in our context.

1 For example, it is reasonable to expect that at one given position the

multipath distance of the sensed signal is likely to remain approx-

imately the same, and in this case a MS which, for instance,

temporarily remains still would take measures affected by similar

NLOS errors—see also the considerations made in [6, 15].

Wireless Netw (2012) 18:579–589 581

123

In the following we will implicitly refer to the mea-

surement model in the case s = 1, and will omit to specify

this.

Also the complexity of the derivation of P(Z1|p*) for a

higher number n of BS in sight can be addressed, from a

practical standpoint, on the basis of the following remark.

When d3� d2� d1 the dominant term in (6) results to be

the first integral, the other terms amounting to corrections

of several order of magnitude smaller.

To realize this, consider, for simplicity, the 2-D case and

assume d2� d1. Let d2 � d1 ¼ D and denote by I1 and I2

the first and second integral term in Eq. 5. Referring to Eqs.

5 and 3, we have in the 2-D case:

I1 ¼ e�k1ðD�z21Þ if z21\D

I1 ¼ e�k2ðz21�DÞ if z21 [ D

whereas I2 ¼ e�k1ðDþz21Þ; 8z21. From these expressions it

can be seen that only for very small values of z21 the values

are comparable (in particular if z21 = 0, I1 = I2) but

for higher values of z21 and especially from values above

z21 ffi D the dominating term is I1, see Fig. 3. I1 is already

one order of magnitude greater than I2 for z21 ¼ lnð10Þ2k1

(considering the most relevant case relative to small values

of z21 which is the case z21\D). In the multidimensional

case a similar reasoning can be followed, according to the

expression (10).

To compute this dominant term, assume that it is pos-

sible to sort the multipath distances di in ascending order,

i.e., such that (with no loss of generality we deal with the

case n = 3, to ease the exposition)

di1\di2\di3 ð7Þ

hence identifying, by this criterion, the three indexes i‘ 2f1; 2; 3g: Now zi2i1 and zi3i1 may be derived as linear

combinations of the actual measurements z21 and z31 (e.g.,

if i1 = 2, i2 = 3, i3 = 1 then zi2i1 ¼ z21 � z31 and zi3i1 ¼z21; etc.). With this choice (and neglecting the wij),

still PðZi1 jp�Þ ¼ PðZ1jp�Þ; but since by construction

jdi‘ � di1 j ¼ di‘ � di1 ; it is possible to easily compute the

dominating term of F2, denoted by F2, as the analogous of

the first integral in the RHS of (6), and then approximate F2

by the normalized F2:

F2 ffi F2 :¼ gZ1

0

fi1ðyÞfi2ðzi2i1 þ yÞfi3ðzi3i1 þ yÞdy; ð8Þ

where g is a normalizing factor. Generalizing to n - 1

TDOA measurements (n BSs), with the convention that

zij = 0, if i = j:

Fn�1 ffi Fn�1 :¼ gZ1

0

Yn

i‘¼1

fi‘ðzi‘i1 þ yÞdy: ð9Þ

According to the condition that identifies the indexes i‘,

the quality of this approximation is not guaranteed

whenever the multipath distances are comparable i.e.,

di % dj, i = j. In any case, unfortunately it is not possible

to check the condition (7), being relative to multipath

distances ‘‘known’’ only through a probabilistic model.

However, once assumed that all readings are taken in

NLOS condition, there is a correlation between (7) and the

condition d�i1\d�i2\d�i3 as a consequence of the fact that the

support of the pdf of P(di|di*) is contained in that of

0 200 400 600 800 1000 12000

2

4

6

8

10

12

x 10−4

meters

P (

z21

| |d

2* −d 1* | =

300

m )

Fig. 2 Measurement model for a single TDOA reading, whose true

value is 300 m/c, neglecting or including a system measurement noise

w21�Nð0; 802Þ (red continuous and black dashed line, respectively).

Here cwnlos�Eð500Þ (Color figure online)

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1F

1

First integral termSecond integral term

Fig. 3 Comparison between F1 (black solid) and F1 (red dash dotted)

and the residual of the approximation, I2 (blue dashed). Here D ¼100 m and k1 = k2 = 1/50 (Color figure online)

582 Wireless Netw (2012) 18:579–589

123

Pðdjjd�j Þ when d�i [ d�j (for example if fi;1 ¼ Eð1=kÞ; 8i, the

probability that d2 [ d1 when d�2 � d�1 ¼ D is

1� 12

expð�2kDÞ; if k = 1/200 this amounts to about 70 %

for D ¼ 100, to about 82 % for D ¼ 200, etc.). For this

reason, as a practical criterion to compute the

approximated measurement model through (9), the set of

indexes i‘ can be identified on the basis of the condition

d�i1\d�i1\ � � �\d�in :

We remark that the proposed approximation rationale is

based on relatively strong positions, that have to be rated

keeping in mind that these falls in the category of working

assumptions, by evaluating, a posteriori, the results achieved

through their exploitation. Here, this will be pursued, in the

next Section, comparing the results achieved by a PF algo-

rithm based either on the approximated measurement model

(8), or on the measurement model (6).

For the exponential distribution of wnlos given in Eq. 3,

denoting f‘ ¼ zi‘i1 � ðd�i‘ � d�i1Þ; kP ¼Q

i ki; kR ¼P

i ki;

and l ¼ maxf0;�f2; . . .;�fng; the approximated pdf of

P(Z1|p*) is:

Fn�1 ¼ gkP

kRexp �

X‘

ki‘f‘ � kRl

" #: ð10Þ

The above approximation is the generalization to an n

dimensional case of the two dimensional result, which is

(after some algebraic computation):

F2 ¼ gZ1

0

f1ðyÞf2ðyþ zÞdy

¼ gk1k2

k1 þ k2

e�k2ðz�ðd2�d1ÞÞ�

e�ðk1þk2Þx���1maxf0;ðd2�d1Þ�zg

�:

To have a glance of the quality of the approximation

Fn�1 ffi Fn�1, we computed, for n = 3, the position

likelihoods (i.e., P(Z1|p*) as a function of p*) given a fixed

pair of readings Z1, reported in Fig. 4. As expected, the

approximation is good for all points far from the region where

the distances from the three bases are comparable, whereas a

worse approximation is achieved near the loci di* - dj

* = 0,

i = j (dashed lines). In Fig. 5 given a point p*, marked by the

asterisk, the relative (ideal) LOS non noisy TDOA readings Z1*

are computed, hence the position likelihoods, P(Z1*|p*), which

are shown. The case reported in Fig. 4 is particularly critical;

notice the better quality of the approximation in the case of

Fig. 5. It it is noteworthy that for z21 and z31 close to zero,

yielding a likelihood P(Z1|p*) which is mostly concentrated

about the point p* characterized by d1* % di

* % d3*, it results

that the approximation is (unexpectedly) very good (the

corresponding figure is not reported for brevity). In all these

cases cwnlos�Eð2d�i Þ, hence considering ki dependent on di*

through the simple relation ki = 1/(.2di*) (a justification for

this choice will be discussed in the next section).

All the simplifying assumptions, despite the approxi-

mations brought, are intended to ease the adoption of

efficient (in terms of computational time) Monte Carlo

localization algorithms, which is the approach pursued

here, implementing a PF, treated in the next Section. A

model available in analytical form, although approximate,

may also facilitate the development of Maximum Likeli-

hood algorithms.

3 Particle filter (PF) localization

The localization is achieved by a Particle filtering proce-

dure, chosen for the non Gaussian measurement model

considered here.

Regarding measurements, real TDOA data were derived

by a method using cross-correlation to estimate both code

BS1

BS2 BS3

−500 0 500 1000 1500 2000 2500

0

500

1000

1500

2000

2500

BS1

BS2 BS3

d*2−d*

1=0

d*3−d*

1=0

−500 0 500 1000 1500 2000 2500

0

500

1000

1500

2000

2500

(a) (b)

Fig. 4 Comparison between F2 and the approximation F2, relative to the readings z21 = 125, z31 = 283. Unit distance is meters. a Position

likelihoods as given by F2, Eq. 6. b Approximated position likelihoods as given by F2 (Eq. 8; here, in particular, Eq. 10)

Wireless Netw (2012) 18:579–589 583

123

and frequency offsets of several BSs at once, where the

transmission system adopts a CDMA technique [7]. The

data are relative to a fixed position, with coordinates (from

now on coordinates and distances are expressed in meters)

p* = [-133, 973], where three BS are visible and located

in [-574, 1,512], [0,0] and [2,657, 121] (BS2 is the origin,

see Figs. 5 and 7).

In the simulation case the scenario is that of a moving

MS, with position p(t) at the tth sampling instant, always in

the range of three BSs (located as in the real measurements

case) and collecting, at each sampling instant, two (noise

corrupted) TDOA readings. The simulated path, see Fig. 7,

comprises 5 segments, followed at different velocities:

10 m/Ts in segments 1 and 4; 5 m/Ts in segment 2, 14.1 m/Ts

in segment 3, and 2 m/Ts in segment 5, with Ts the sam-

pling interval; the MS is stopped for 40 sampling instants

(hence having null velocity) at point [1,000, 1,000]. Here

and in the following, Ts has been chosen conventionally

equal to one time unit.

To generate the artificial readings, three Markov Chains,

with state si 2 f0; 1g, model the dynamics of the sight con-

ditions between MS and each BS (the transition matrices are

characterized by paa = .9 and pab = .1, a, b = 0, 1). At

each sampling instant, di(t) is computed according to the

actual true distance di*(t), the current value of si(t) and a

sample of wnlos, by Eq. 2.

For the implementation of the simulations presented here,

we chose cwnlos�Eðad�i Þ, with a a parameter by which the

level of additive NLOS error can be tuned (in the experi-

ments that will follow we tested the algorithms for a ranging

from a! 0, corresponding to the LOS conditions, up to

a = .3, which corresponds to severe NLOS conditions). An

example of multipath distances generated in this fashion is

illustrated in Fig. 6, for a = .2. This choice for the ki’s is

motivated by the fact that, according to [3], the lag due to

multipath is dependent on the LOS distance. In particular,

taking ki inversely proportional to di* corresponds to assume

that the expected lag due to multipath is proportional to the

real distance, or, in other words, that the multipath scenario

visualized on Fig. 1 does not depend on the scale of such

figure. The one used here is not intended as a unique model of

the NLOS conditions. Other models may be well suited as

well. As for our own tests, other results, not reported here for

brevity, relative to values of ki independent by di* and con-

stant, do not change the essence of the performance results;

indeed, the validity of the general model (6) and his

n dimensional extension Fn-1 does not require any precise

choice for what concerns the ki.

Having sampled wnlos as described above, then z21(t) and

z31(t) are generated, by (4), according to samples of w21

BS1

BS2 BS3

−1000 −500 0 500 1000 1500 2000 2500 3000

−500

0

500

1000

1500

2000

2500

BS1

BS2 BS3

d2* −d

1* =0

−1000 −500 0 500 1000 1500 2000 2500 3000

−500

0

500

1000

1500

2000

2500

(a) (b)

Fig. 5 Comparison between F2 and the approximation F2, relative to

hypothetical LOS non noisy TDOA readings from the point p* =

[ -133, 973] (marked by the asterisk) i.e., zi* = 285, z31

* = 2,220.

The plots show P(z21* , z31

* |p) as a function of p. Unit distance is meters.

a Position likelihoods as given by F2, Eq. 6. b Approximated position

likelihoods as given by F2 (Eq. 8; here, in particular, Eq. 10)

0 50 100 150 200 250 300 350 400500

1000

1500

2000

2500

3000

3500

4000

4500

meters

LOS

and

NLO

S d

ista

nces

Fig. 6 True and multipath distances from BS1 in the simulated path

(i.e., d1* and d1 of Eq. 1) versus sampling instants

584 Wireless Netw (2012) 18:579–589

123

and w31 distributed as Nð0; 802Þ (Ref. [15] reports for rlos

values of the order of 60–100m). From now on we will

always consider the pdf Nð0; 802Þ for the generation of

wlos, whereas the parameters characterizing wnlos will be

varied, according to the type of test to be conducted.

A standard motion model is adopted, with 4 states rep-

resenting two (planar) position coordinates and velocities,

i.e., x(t ? 1) = x(t) ? vx(t)Ts ? xx(t)Ts2/2 and vx(t ? 1) =

vx(t) ? xx(t)Ts, with xxð�Þ the process noise (similar

dynamics hold for y and vy, with xxð�Þ and xyð�Þindependent).

The PF implementation is also quite standard, see e.g.,

[23]. To sketch the algorithm, let nk(t) and wk(t) denote,

respectively, the coordinates (n = [x, y]) and the weight of

particle k at time t; k 2 f1; 2; . . .;Mpg;Mp the number of

particles (In all the experiments we used Mp = 2500).

Algorithm 1 (PF) The particles are initialized sampling

uniformly in a disk region comprising all BSs (e.g., their

circumcircle). At each step t 2 f1; 2; . . .; Tg do:

• Apply the motion model defined above to each particle,

which gives nk(t ? 1) as a function of nk(t) and the

current samples xxk(t), xy

k(t).

• Apply the measurement model to each particle and

normalize the weights (with constant g) so that

wk(t ? 1) = gP(Z1(t ? 1)|nk(t ? 1)).

• Resample (we use a low variance resampling proce-

dure, see e.g., [23]). Estimate the position as the mean

of the new (resampled) set of particles.

3.1 Simulation test: a = .2

As a first simulation test, we present the results of one run

of the PF algorithm for which the readings are artificially

generated according to the choice a = .2 (see Fig. 6). In

this first example the measurement model adopted is given

by F2, according to the pdf’s (3) and taking the same values

for the ki’s which were adopted to generate the artificial

readings i.e., ki ¼ 1=ð:2diÞ, where di are the estimated

distances from MS to the BSi. Hence, in this case we use

the complete model, without any approximation. The

results are illustrated in Fig. 7, showing the results of the

position estimation. Convergence is typically achieved

quite quickly (after 20–30 steps; see Fig. 8 that represents a

typical observed behavior for the performance evolution):

as a measure of the estimation performance we chose the

mean and the standard deviation of the distance error from

step #50 to the end, denoted by J and Jr, respectively.

The performances, rounded to the nearest integer, are

J = 93 and Jr = 51. Adopting the approximated model F2;

and with the same artificial readings, one PF run achieves

J = 97 and Jr = 53.

To compare the effects of the adoption of the two ver-

sion of the measurement model (i.e., both the exact and the

approximated one), these performances were evaluated for

20 Monte Carlo simulations, in which were generated

different artificial measures Z1 according to the procedure

described above. The average of the J performance mea-

sure was �J ¼ 88:8 adopting the exact model F2 and �J ¼90:8 adopting the approximated model F2; the averaged Jr

are 53.6 and 52.8, respectively. The fluctuation of J about

the mean �J is characterized, in the two cases, by a standard

deviation equal to 10.4 and 8.8.

Remark 2 In our simulation tests, xx and xy of the PF

were sampled from a stationary Gaussian distribution with

zero mean and standard deviation rx = 1.5. As it can be

expected, if the path were followed at a constant velocity,

−600 −400 −200 0 200 400 600 800 1000 1200

0

200

400

600

800

1000

1200

1400

1600 BS1

BS2

Seg.1 Seg.2

Seg.3Seg.4

Seg.5

Fig. 7 Estimated versus true positions in the simulated path (first 25

positions estimates are not depicted, as well as BS3, which is outside

the border)

0 50 100 150 200 250 300 350 4000

500

1000

1500di

stan

ce e

rror

50 100 150 200 250 300 350 4000

50

100

150

200

250

sampling instants

dist

ance

err

or

Fig. 8 Distances between true and estimated positions for one PF

estimation run relative to the simulation case

Wireless Netw (2012) 18:579–589 585

123

by properly tuning the constant parameter rx better per-

formances could be achieved; in our case the choice of a

multiple velocity path makes the scenario more proba-

tionary, as a possible tuning of rx would however result in

the choice of a fixed value for it, which could not fit for all

the segments of the path. Following another artificial path

characterized by constant velocity 10 m/Ts, we obtain, after

20 Monte Carlo simulations, �J ¼ 66:2 and �J ¼ 70:0

adopting the exact model F2 and the approximated one F2,

respectively. The fluctuation of �J about the mean is char-

acterized, in the two cases, by a standard deviation equal to

9.7 and 5.3.

Remark 3 Performances slightly improve if the position is

estimated before resampling is performed, since resam-

pling usually increases the variance of the estimates.2

Trying this different estimation scheme we obtained,

adopting the model F2; �J ¼ 86:1 3 8:6, whereas the

performances reported above, relative to the standard PF

version are: �J ¼ 90:8 3 8:8.

3.2 Simulation test: a 2 ð0; 0:3�

To test further the algorithm, we conducted several simu-

lation tests with various values of a 2 ð0; 0:3�, i.e, from

LOS conditions (a! 0) to severe NLOS conditions

(a = .3). Now, in realistic setting one could only guess the

covariance related to the observations, and could tune an

algorithm by choosing a fixed value for its parameters. For

this reason, our PF adopted the approximated measurement

model F2, with the pdf’s in Eq. 3 always with the fixed

choice ki ¼ 1=:15di (i.e., assuming always a = .15), while

the simulated measurements have been generated accord-

ing to the various values of a from a! 0 up to a = .3.

On the other hand, a standard extended Kalman filter

(EKF) was also set up for position estimation. The covari-

ance matrices of the EKF were tuned as follows, depending

on the LOS versus NLOS scenario. In the LOS case the

matrix R, relative to the measurement (observations) noise

covariance was set: R ¼ diagð802; 802Þ, according to the

gaussian LOS additive noise generated sampling from

Nð0; 802Þ. Then, with this fixed value of R, we tuned the Q

matrix (relative to the noise in the dynamics) by searching in

the two dimensional space of diagonal matrices

Q ¼ diagðr2s ; r

2s ; r

2v ; r

2vÞ, with the objective of minimizing

the index J mentioned above. We finally obtained

Q ¼ diagð1; 1; 42; 42Þ: notice that rv = 4 reflects the stan-

dard deviation relative to the velocities of our artificial path.

For all the NLOS cases, hence characterized by non zero

values for a, we kept fixed the Q matrix whereas we set

R ¼ diagðr2z ; r

2z ; Þ, where r2

z are the computed sample

variances of the noise actually affecting the artificial

readings adopted. We remark that this is possible in the

simulative cases, since, having artificially generated the

readings, had the chance to compute the noise variances.

The performances of such an ‘‘artificial’’ best possible EKF

resulted in a kind of ‘‘lower bound’’ in the performances

and were used as a comparison term.

Despite this, nevertheless sometimes even this ‘‘artifi-

cial’’ version of EKF generates estimated position diverg-

ing from the true one, especially for severe NLOS

conditions. The reason lies in the fact that the EKF is a

local method, and adopts a zero mean gaussian hypothesis

for the additive error; hence, when in presence of high

NLOS noise, it could happen that for several samples the

estimated position drifts from the true one of an amount

that makes it impossible for this method to recover the true

trajectory, since, after such drift, only local corrections are

possible, due to the linearization inherent to the EKF

algorithm.

For each value of a we run 20 EKF simulations and 20

PF simulations. According to the analysis of the results, we

introduced the criterion J [ 500 as an heuristic to discern

whether the algorithm achieved convergence or not. Sub-

sequently, for each value of a we computed the percentage

of non converging simulations, and the average of the

J performances only for the convergent simulations. The

results are summarized in Figs. 9 and 10.

The role of the assumption of a diagonal structure for

the measurements noise covariance matrix R has been

investigated in the simulated scenario by considering also a

non diagonal case. In particular, the same 4-D EKF sce-

nario considered above has been simulated, with the same

measurement noise and same values for the Q matrix, but

with a R matrix having off-diagonal elements equal to half

0 0.05 0.1 0.15 0.2 0.25 0.30

5

10

15

20

25

30

35

% n

on c

onve

rgin

g si

mul

atio

ns

α parameter

EKF

PF

Fig. 9 Percentage of non converging EKF and PF runs2 We thank an anonymous reviewer for this hint.

586 Wireless Netw (2012) 18:579–589

123

of the diagonal ones. The difference between the perfor-

mances (J index) in the two cases is reported in Fig. 11.

3.3 Experimental test

A comparison between EKF and PF could be performed

with real data, relative to a fixed MS position, see Fig. 5, in

an urban scenario. As these readings are relative to a fixed

position, a better EKF estimate was achieved by the

adoption (only for the EKF) of a ‘‘degenerate’’ 2-D motion

model of the kind x(t ? 1) = x(t) ? xx(t)Ts (and analo-

gously for the y coordinate). Being the EKF a ‘‘local’’

method, the initial condition for the EKF position estimate

was set equal to the true position perturbed by an error

Nð0; 102Þ, for each coordinate. In addition, also a 4 state

variables EKF is considered, based on the same motion

model used in the previous simulation tests (i.e., the model

taking into account both position and velocity). The initial

condition for such a 4-D EKF have been chosen similarly

to the above 2-D case. The covariance matrices for the 2-D

EKF have been chosen by exhaustive search over the space

of matrices Q ¼ diagðr2q; r

2qÞ and R ¼ diagðr2

r ; r2r Þ, with

the objective of minimizing the estimation error over all

the data sets. This yielded the choice Q ¼ diagð1; 1Þ and

R ¼ diagð1002; 1002Þ, which is in line with actual

measurement errors. The matrices for the 4-D EKF has

been tuned with a similar approach, achieving: Q ¼diagð1; 1; 1; 1Þ and R ¼ diagð1002; 1002Þ.

For the sake of comparison, we also report the data for a

2-D PF, which has been tuned by an approach similar to the

above for the EKF’s: an exhaustive search over the space

of variance values for the underlying brownian motions.

Such an approach is clearly not feasible in a real scenario

and is only intended for comparison.

It is stressed that such a tuning procedure is only pos-

sible in those experimental settings (such as our our case),

where true MS position is known in advance. Hence, the

results achieved trough the two EKF’s represent a kind of

lower bound on estimation error.

The PF estimate was achieved by the same motion

model described in Algorithm 1 and with no prior infor-

mation about the true position, hence initializing the par-

ticles as described in Algorithm 1.

The various data set obtained from actual reading of

TDOA were analyzed in terms of the presence of NLOS

readings and ordered accordingly to an increasing amount

of NLOS readings: from Set 1 which comprises only LOS

readings, up to Set 9 which contains almost 50% of NLOS

measurements. Hence, going from Sets 1 to 9 with real data

0 0.05 0.1 0.15 0.2 0.25 0.30

50

100

150

200

250

300

350

α parameter

perf

orm

ance

J EKF

PF

Fig. 10 Performances of the converging EKF and PF executions

(mean of the J0s obtained in convergent executions)

0 0.05 0.1 0.15 0.2 0.25 0.3−10

−5

0

5

10

15

20

α parameter

Var

iatio

n on

per

form

ance

J

Fig. 11 Difference on performances between the converging EKF for

non diagonal R and diagonal one

1 2 3 4 5 6 7 8 920

30

40

50

60

70

80

90

100

110

120

Data Set #

perf

orm

ance

J

2−dim. EKF

4−dim. EKF

4−dim PF

2−dim PF

Fig. 12 Comparison of 2-, 4-D EKF, 4- and 2-D PF for real data set

with increasing NLOS noise

Wireless Netw (2012) 18:579–589 587

123

is equivalent to increasing parameter a from zero to larger

values in the simulation tests.

The experimental results indicate that the PF localiza-

tion method is more robust and achieves acceptable per-

formances in all the cases considered. Also, notice that PF

has a performance which is alway better than the 4-D EKF.

PF also outperforms the 2-D EKF, with the only exception

of two data set (sets 2 and 4) corresponding to a reduced

level of NLOS noise. At the same time, in case of large

NLOS noise, the PF is remarkably better than the EKF, and

the two EKF considered yield similar results.

Notice that the relative performances of PF and EKF and

their dependence on NLOS noise level in such a real data

scenario are of the same kind revealed by the simulation

tests.

We also remark that, for the two EKF used as a com-

parison, both the initialization strategy and the choice of

the motion model give an advantage to this method, not

exploited by the PF.

The role of the assumption of a diagonal structure for

the measurements noise covariance matrix R (setting a

positive value for the off diagonal elements is a conse-

quence of the possible correlation among data, due to the

common base station) has been investigated also in this

case. Results indicate a difference on performance on the

order of 1% between an EKF with non diagonal matrix

and and EKF with diagonal one, keeping fixed the

Q matrix.

4 Conclusion

We addressed the problem of TDOA localization by

means of a PF procedure. The solution relies on the

development of a general measurement model which

accounts for the correlation among the measurements due

to the common reference base; to our knowledge this has

not been presented in the literature. This general model

could be particularized for any standing assumption for

the distributions of LOS/NLOS readings. Here, this model

has been adopted for the implementation of a PF esti-

mating only the position of the MS; as such, several

working assumption, discussed in the paper, have been

adopted in order to keep the complexity of the measure-

ment model tractable. The experiments were conducted

for an exponential model of NLOS errors; a choice based

on conjectures drawn from the literature, and also

accounting for the computational complexity of the

resulting model. The results showed the feasibility of this

approach and the possibility to employ the simplifying

working assumptions.

5 Future work

The paper can be further extended along several directions.

A first extensions that we feel of considerable interest, and

mentioned in Remark 1, is that of conducting measurement

campaigns and modeling activities to study NLOS error

distribution taking into account the true distance from BS,

i.e., to study the conditioned probability of NLOS noise.

And further, it appears quite important to also take into

account the autocorrelation of the NLOS noise.

Another important extension, which we are currently

planning to afford, concerns the study of theoretical lower

bound on estimation errors. Other directions of future

research include the integration of TDOA measurements

with measurement from GNSS (global navigation satellite

system), as well as the use of a similar approach in indoor

positioning based on UWB techniques.

Finally, it is mentioned that the EKF filters could be also

‘‘tuned’’ by incorporating an a-priori known mean for the

measurements. Such an issue will be further extended in

future work.

Acknowledgments The authors would like to thank Prof. G. Lachapelle,

University of Calgary, for the hardware used to receive and collect

IS-95 data.

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Author Biographies

Mauro Boccadoro was born in

1972, received the Laurea degree

magna cum laude in Electronic

Engineering from the University

of Ancona in 1999 and the Ph.D.

in Information Engineering from

the University of Perugia in

2005, Italy. In the period

2000–2002 he was at the MiTech

Lab of Scuola Superiore

Sant’Anna, Pisa, Italy, as a

research assistant; in 2003 he was

a visiting student at Georgia

Tech, Atlanta, GA, USA. He is

currently Assistant Professor at

DIEI, University of Perugia. His research interests include optimal

control and estimation, manufacturing systems, hybrid systems, and

systems biology.

Guido De Angelis graduated in

electronics engineering in

1993 at Universita degli Studi di

Perugia (University of Perugia).

He discussed a final experimen-

tal dissertation on high-fre-

quency receiver of Itelco S.p.A.,

Orvieto. Ph.D. in Electronic

Engineering (Telecommunica-

tions) in 2011 at the Department

of Electronic and Information

Engineering of the University of

Perugia. Argument: ‘‘Study and

integration of ground-based and

satellite-based positioning sys-

tems’’. He is studying satellite navigation and in particular: weak signal

and aided global positioning system (AGPS), both in GPS and Galileo

Systems. He also works at Regione Umbria (Regional Government of

Umbria), Office for innovation promotion and enterprise innovation

services, where he is also responsible for designing and implementing

the information technology network infrastructure.

Paolo Valigi was born in 1961.

He received the Laurea degree in

1986 from University of Rome

La Sapienza and the Ph.D.

degree from University of Rome

Tor Vergata in 1991. He was

with Fondazione Ugo Bordoni

from 1990 to 1994. From 1994 to

1998 he was research assistant at

University of Rome Tor Vergata.

From 1998 to 2004 he has been

associate professor at University

of Perugia, where since 2004

he is full professor of System

Theory, with the Department of

Electronics and Informatics Engineering. His research interests are in

the field of systems biology, robotics, and distributed control and

optimization. He has authored or co-authored more than 100 journal and

conference papers and book chapters.

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